# CMB spectra and likelihood code

## Contents

## 2018 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 ℓ = 2-2508. Over the multipole range ℓ = 2-29, the power spectrum is derived from the "Commander" component-separation algorithm applied to the combination of Planck 2018 temperature data between 30 and 857 GHz, including 86% of the sky (Planck-2020-A6^{[1]}). The asymmetric error bars associated with this spectrum are the 68% confidence limits and include the uncertainties due to foreground subtraction.

For multipoles equal or greater than ℓ = 30, instead, the spectrum is derived from the "Plik" cross-half-mission likelihood Planck-2020-A5^{[2]}, with foreground and other nuisance parameters fixed to a best fit assuming the base-ΛCDM cosmology. Associated 1σ errors include beam uncertainties. Commander is described in more detail in Planck-2020-A4^{[3]}, and Plik is described in more detail in Planck-2020-A5^{[2]}.

#### TE, EE, and EB, BB[edit]

The Planck best-fit CMB polarization and temperature-polarization cross-correlation power spectra, shown in the figure below, cover the multipole range ℓ = 2-1996. In the multipole range 2 ≤ l ≤ 29, we plot the power spectra estimates from the SimAll likelihood (though only the EE spectrum is used in the baseline parameter analysis at l ≤ 29), see Planck-2020-A6^{[1]}). The spectra are obtained using a cross quasi maximum likelihood algorithm (QML) on the 100 and 143 GHz maps and validated against high fidelity end-to-end simulations.

In the range ℓ = 2-29, we also release the *BB*, and *EB* power spectra derived from the same maps. Error bars are given as the 68% confidence intervals as derived from the Fisher information matrix of the estimates.
Analogously to the *TT* case, the ℓ ≥ 30 spectrum is derived from the Plik likelihood Planck-2020-A3^{[4]} 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 ΛCDM run.

#### Best-fit model[edit]

We also provide best-fit LCDM CMB power spectra from the baseline Planck TT,TE,EE+lowE+lensing. The spectra must be divided by the best-fit Planck map-based calibration parameter squared, calPlanck**2, to be compared to the coadded CMB spectra. The best-fit calPlanck value can be found in the file "COM_PowerSpect_CMB-base-plikHM_TTTEEE-lowl-lowE-lensing-minimum_R3.01.txt".

### Production process[edit]

The Plik high-multipole likelihood (described in detail in Planck-2020-A5^{[2]}) is a Gaussian approximation of the probability distributions of the *TT*, *EE*, and *TE* angular power spectra, with semi-analytic covariance matrices calculated assuming a fiducial cosmology. It includes multipoles in the range 30 to 2508 for TT and 30 to 1996 for *TE* and *EE* and is constructed from half-mission cross-spectra measured from the 100-, 143-, and 217-GHz HFI frequency maps. For more details see Planck-2020-A6^{[1]}.

### Inputs[edit]

The T T likelihood uses four half-mission cross-spectra with different multipole cuts to avoid multipole regions where noise dominates due to the limited resolution of the beams and to en-sure foreground contamination is correctly handled by our fore-ground model: 100 × 100 ( ℓ = 30–1197); 143 × 143 (ℓ = 30– 1996); 143 × 217 (ℓ = 30–2508); and 217 × 217 (ℓ = 30–2508). The TE and EE likelihoods also include the 100 × 143 and 100 × 217 cross-spectra to improve the signal-to-noise ratio, and have different multipole cuts: 100 × 100 (ℓ = 30–999); 100 × 143 (ℓ = 30–999); 100 × 217 (ℓ = 505–999); 143 × 143 (ℓ = 30– 1996); 143 × 217 (ℓ = 505–1996); and 217 × 217 (ℓ = 505– 1996). The 353, 545 and 857 GHz temperature and polarization maps are also used to build the dust templates, and contribute to the galactic and point source masks determination.

### File names and meta-data[edit]

The CMB spectra and their uncertainties are distributed in ASCII text files named *COM_PowerSpect_CMB_nn_R3.01.fits*, where nn stands for the type of spectrum the file contains:

- COM_PowerSpect_CMB-EE-full_R3.01.txt
- COM_PowerSpect_CMB-TE-full_R3.01.txt
- COM_PowerSpect_CMB-TT-full_R3.01.txt
- COM_PowerSpect_CMB-low-ell-BB-full_R3.01.txt
- COM_PowerSpect_CMB-low-ell-EB-full_R3.01.txt

In addition we provide one file containing all the parameters of the Plik runs which yielded the spectra. This file is named

- COM_PowerSpect_CMB-base-plikHM_TTTEEE-lowl-lowE-lensing-minimum_R3.01.txt

The theoretical spectrum of the best-fit model itself is provided in a separate file named

- COM_PowerSpect_CMB-base-plikHM_TTTEEE-lowl-lowE-lensing-minimum-theory_R3.01.txt

The data file columns give *D*_{ℓ} = ℓ(ℓ+1)*C*_{ℓ} / 2π in units of μK^{2}, and the lower and upper 68% uncertainties.

## 2018 Likelihood[edit]

The 2018 Likelihood code will be released at a later time.

## Previous Releases: (2015) and (2013) CMB spectrum and Likelihood[edit]

**2015 CMB spectrum and Likelihood**

**2015 CMB spectra**

**General description**

**TT**
The Planck best-fit CMB temperature power spectrum, shown in the figure below, covers the wide range of multipoles ℓ = 2-2508. Over the multipole range ℓ = 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^{[5]}). The asymmetric error bars associated with this spectrum are the 68% confidence limits and include the uncertainties due to foreground subtraction.

For multipoles equal or greater than ℓ = 30, instead, the spectrum is derived from the "Plik" likelihood Planck-2015-A11^{[6]} 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 ΛCDM run. Associated 1σ errors include beam uncertainties. Both Commander and Plik are described in more detail in the sections below.

**TE, EE, and TB, EB, BB **

The Planck best-fit CMB polarization and temperature-polarization cross-correlation power spectra, shown in the figure below, cover the multipole range ℓ = 2-1996. The data points relative to the multipole range ℓ = 2-29 are quadratic maximum likelihood (QML) estimates from foreground-cleaned Planck 70-GHz *Q* and *U* Stokes parameter maps using 46% of the sky (the same maps that are used in the "lowP" likelihood, see Planck-2015-A11^{[6]}). In the range ℓ = 2-29, we also release the *BB*, *TB*, and *EB* power spectra derived from the same maps (for the cross-spectra involving temperature, the Commander map is always used). Symmetric error bars are given as the 68% confidence intervals as derived from the Fisher information matrix of the estimates.
Analogously to the *TT* case, the ℓ ≥ 30 spectrum is derived from the Plik likelihood Planck-2015-A11^{[6]} 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 ΛCDM run.

**Production process**
The ℓ < 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 parameters (Planck-2015-A10^{[5]}). The power spectrum at any multipole ℓ is given as the maximum probability point for the posterior *C*_{ℓ} distribution, marginalized over the other multipoles, and the error bars are from the 68% confidence level; see Planck-2015-A10^{[5]}. The polarization spectra (*EE*, *TE*, *BB*, *TB*, *EB*) are estimated from maps of the Planck 70 GHz channel, cleaned from polarized synchrotron emission using the 30 GHz channel as a template and from polarized dust emission using the 353 GHz channel; see Sect.2 of Planck-2015-A11^{[6]} for details about the foreground cleaning procedure. The pixel-pixel (map level) noise covariance matrix has been estimated from the timelines and corrected to account for foreground cleaning. The QML code Bolpol has been used to estimate the power spectra and their Fisher matrix from the cleaned map and covariance matrix, after imposing a Galactic mask that makes 43% of the full sky available to analysis (see again Planck-2015-A11^{[6]}). The symmetric 68% error bars have been derived from the resulting Fisher matrix and include contributions from cosmic variance, noise, and foreground associated errors.

The ℓ ≥ 30 part of the *TT*, *TE*, and *EE* power spectra have been derived by the Plik likelihood, a code that implements a pseudo-*C*_{ℓ} based technique, extensively described in section 2.2 and the appendix of Planck-2013-XV^{[8]}, and more recently in Planck-2015-A11^{[6]}. 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^{[7]} and in Planck-2015-A11^{[6]}. The final power spectrum is an optimal combination of the 100, 143, 143×217, 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^{[7]}). A thorough description of the models of unresolved foregrounds is given in Planck-2015-A11^{[6]}. The spectrum covariance matrix accounts for cosmic variance and noise contributions, together with beam uncertainties. The ℓ ≥ 30 CMB *TT* spectrum and associated covariance matrix are available in two formats.

- Unbinned: TT, 2479 bandpowers (ℓ = 30-2508); TE or EE, 1697 bandpowers (ℓ = 30-1996).
- Binned, in bins of Δℓ = 30: TT, 83 bandpowers; TE or EE, 66 bandpowers.

We bin the *C*_{ℓ} power spectrum with a weight proportional to ℓ(ℓ+1), so that the *C*_{ℓb} binned bandpower centred on ℓ_{b} is
Equivalently, using the matrix formalism, we can construct the binning matrix *B* as
where *B* is an *n*_{b}×*n*_{ℓ} matrix, with *n _{b}*=83 being the number of bins and

*n*

_{ℓ}=2479 the number of unbinned multipoles. Thus Here, is the vector containing all the binned (unbinned)

*C*

_{ℓ}bandpowers, is the covariance matrix, and ℓ

_{b}is the weighted average multipole in each bin. Note that following this definition, ℓ

_{b}can be a non-integer. The binned

*D*

_{ℓb}power spectrum is then calculated as

**Inputs**

- Low-ℓ spectrum (ℓ<30):

- 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 of Haslam et al. (1982);
- Commander χ
^{2}-based LM93 confidence mask (Planck-2015-A10^{[5]}).

- High-ℓ spectrum (30≤ℓ≤2508):

- 100, 143, 143×217 and 217 GHz spectra and their covariance matrix (setion 3.3 of Planck-2015-A13
^{[7]}); - best-fit foreground templates and inter-frequency calibration factors (table 3 of Planck-2015-A13
^{[7]}); - beam transfer function uncertainties (Planck-2015-A07
^{[9]}).

** File names and meta-data **

The CMB spectra and their uncertainties are distributed in a single multi-extension FITS file named *COM_PowerSpect_CMB_R2.nn.fits*.

- R2.00 contains (unbinned) TT spectra for low ℓ and TT, TE and EE spectra at high ℓ, both binned and unbinned (7 extensions).
- R2.01 corrects a small error in the
*effective*ℓ of the bin of the binned data, which was truncated to an integer. Since these are weighted averages of the ℓ's used in a particular bin, they should be a reals. - R2.02 contains low ℓ *E and *B spectra in addition to the TT spectra (5 additional extensions for a total of 12 extensions).

Further details on the data columns are given below (the extension numbers correspond to the R2.02 release).

- 1. TT low-ℓ, unbinned (TTLOLUNB)
- with the low-ℓ part of the spectrum, not binned, and for ℓ=2-29. The table columns are:

*ELL*(integer), multipole number;*D_ELL*(float),*D*_{ℓ}as described above;*ERRUP*(float), the upward uncertainty;*ERRDOWN*(float), the downward uncertainty.

- 2. TE low-ℓ, unbinned (TTLOLUNB)
- with the low-ℓ part of the spectrum, not binned, and for ℓ=2-29. The table columns are:

*ELL*(integer), multipole number;*D_ELL*(float),*D*_{ℓ}as described above;*ERRUP*(float), the upward uncertainty;*ERRDOWN*(float), the downward uncertainty.

- 3. EE low-ℓ, unbinned (TTLOLUNB)
- with the low-ℓ part of the spectrum, not binned, and for ℓ=2-29. The table columns are:

*ELL*(integer), multipole number;*D_ELL*(float),*D*_{ℓ}as described above;*ERRUP*(float), the upward uncertainty;*ERRDOWN*(float), the downward uncertainty.

- 4. TB low-ℓ, unbinned (TTLOLUNB)
- with the low-ℓ part of the spectrum, not binned, and for ℓ=2-29. The table columns are:

*ELL*(integer), multipole number;*D_ELL*(float),*D*_{ℓ}as described above;*ERRUP*(float), the upward uncertainty;*ERRDOWN*(float), the downward uncertainty.

- 5. EB low-ℓ, unbinned (TTLOLUNB)
- with the low-ℓ part of the spectrum, not binned, and for ℓ=2-29. The table columns are:

*ELL*(integer), multipole number;*D_ELL*(float),*D*_{ℓ}as described above;*ERRUP*(float), the upward uncertainty;*ERRDOWN*(float), the downward uncertainty.

- 6. BB low-ℓ, unbinned (TTLOLUNB)
- with the low-ℓ part of the spectrum, not binned, and for ℓ=2-29. The table columns are:

*ELL*(integer), multipole number;*D_ELL*(float),*D*_{ℓ}as described above;*ERRUP*(float), the upward uncertainty;*ERRDOWN*(float), the downward uncertainty.

- 7. TT high-ℓ, binned (TTHILBIN)
- with the high-ℓ part of the spectrum, binned into 83 bins covering 〈ℓ〉= 47-2499 in bins of width ℓ=30 (with the exception of the last bin that is smaller). The table columns are:

*ELL*(float), mean multipole number of bin;*L_MIN*(integer), lowest multipole of bin;*L_MAX*(integer), highest multipole of bin;*D_ELL*(float),*D*_{ℓ}as described above;*ERR*(float), the uncertainty.

- 8. TT high-ℓ unbinned (TTHILUNB)
- with the high-ℓ part of the spectrum, unbinned, in 2979 bins covering 〈ℓ〉= 30-2508. The table columns are:

*ELL*(integer), multipole;*D_ELL*(float),*D*_{ℓ}as described above;*ERR*(float), the uncertainty.

- 9. TE high-ℓ, binned (TEHILBIN)
- with the high-ℓ part of the spectrum, binned into 83 bins covering 〈ℓ〉= 47-1988 in bins of width ℓ=30 (with the exception of the last bin that is smaller). The table columns are:

*ELL*(float), mean multipole number of bin;*L_MIN*(integer), lowest multipole of bin;*L_MAX*(integer), highest multipole of bin;*D_ELL*(float),*D*_{ℓ}as described above;*ERR*(float), the uncertainty.

- 10. TE high-ℓ, unbinned (TEHILUNB)
- with the high-ℓ part of the spectrum, unbinned, in 2979 bins covering 〈ℓ〉= 30-1996. The table columns are:

*ELL*(integer), multipole;*D_ELL*(float),*D*_{ℓ}as described above;*ERR*(float), the uncertainty.

- 11. EE high-ℓ, binned (EEHILBIN)
- with the high-ℓ part of the spectrum, binned into 83 bins covering 〈ℓ〉= 47-1988 in bins of width ℓ=30 (with the exception of the last bin that is smaller). The table columns are:

*ELL*(float), mean multipole number of bin;*L_MIN*(integer), lowest multipole of bin;*L_MAX*(integer), highest multipole of bin;*D_ELL*(float),*D*_{ℓ}as described above;*ERR*(float), the uncertainty.

- 12. EE high-ℓ, unbinned (EEHILUNB)
- with the high-ℓ part of the spectrum, unbinned, in 2979 bins covering 〈ℓ〉= 30-1996. The table columns are:

*ELL*(integer), multipole;*D_ELL*(float),*D*_{ℓ}as described above;*ERR*(float), the uncertainty.

The spectra give *D*_{ℓ} = ℓ(ℓ+1)*C*_{ℓ} / 2π in units of μK^{2}. The covariance matrices of the spectra will be released at a later time.

The CMB spectra are also given in seven simple text files, corresponding to each of the FITS file BINTABLE extensions described above.

**Likelihood**

The 2015 baseline likelihood release consists of a code package and a single data package. Four extended data packages are also available enabling exploration of alternatives to the baseline results. The code compiles to a library allowing for the computation of log likelihoods for a given data set. Each data package contains multiple data sets. A data set permits the computation of a single likelihood among:

- the high-ℓ temperature and polarization CMB (jointly or separately);
- the low-ℓ temperature and polarization CMB (jointly or separately);
- the CMB lensing reconstruction.

By combining the results from different data sets (possibly from different data packages), one can compute the likelihood of different subsets of the Planck data.

Only the baseline data package, *COM_Likelihood_Data-baseline_R2.00.tar.gz*, will be fully described on this page. It contains six data sets, which are enough to compute all of the baseline Planck results that are discussed in Planck-2015-A12^{[10]}. In particular it allows for the computation of the CMB and lensing likelihood from either the Temperature data only, or the Temperature + Polarization combination.

The other data packages contain data sets that extend the baseline results, enabling the exploration of different regimes, which are discussed in Planck-2015-A11^{[6]} and Planck-2015-A15^{[11]}. Their full description is contained in the documentation included in each of the extended data packages.

**Library and tools**

**Description**
The library consists of code written in C and Fortran 90. It can be called from both of
those languages. Optionally, a python wrapper can be built as well. Scripts to
simplify the linking of the library with other codes are part of the package, as
well as some example codes that can be used to test the correct installation of
the code and the integrity of the data packages. Optionally, a script is also available, allowing
the user to modify the multipole range of the TT likelihoods and reproduce the hybridization
test performed in the paper.
A description of the tool, the API of the library, as well as different
installation procedure are detailed in the *readme.md* file in the code package.

**File name**
The code can be extracted from
*COM_Likelihood_Code-v2.0_R2.00.tar.bz2*.

Please read the file *readme.md* for installation instructions.

**Data sets - Baseline data**
All of the released baseline data are distributed within a single file,
*COM_Likelihood_Data-baseline_R2.00.tar.gz*.

This file extracts to a directory hierarchy, containing the different data sets needed to compute different likelihoods.

Each data set is stored in its own directory. The directory structure follows similar rules for each data set, and stores data, template, and meta-data for each particular likelihood.

As in 2013, the CMB likelihood is cut into low-ℓ and high-ℓ parts. Moreover,
for each of those, we distribute both a T-only datafile and a joint T+P one. In the case
of the high-ℓ part, we also distribute two specific versions of the likelihood,
that allow for estimate of T and T+P marginalized over the nuisance parameters.
Combining both the low-ℓ and high-ℓ files, one can compute the likelihood over the
range ℓ=2-2508 in TT and ℓ=2-1996 in TE and EE. A full description of the low-ℓ
and high-ℓ likelihood is available in Planck-2015-A11^{[6]}.

Similarly, for the case of lensing, we also distribute the likelihood using both
the reconstruction based on the *T* map only, or on *T*+*P* maps.
A full description of the lensing likelihood is available in Planck-2015-A15^{[11]}.

All of the likelihood files have at least one nuisance parameter, allowing users to investigate the Planck absolute calibration. We recommend that his parameter is explored in the Gaussian prior 1.0000±0.0025.

** Low-ℓ likelihoods **

**TT only - commander**
This file allows for the computation of the CMB TT likelihood in the range ℓ=2-29.
Using the optional tool in the code package, it can be modified to cover any multipole range
up to ℓ<200.

** Production process**
The likelihood is based on the results of the Commander approach, which implements a
Bayesian component separation method in pixel space, sampling the posterior distribution
of the parameters of a model that describes both the CMB and the foreground emissions
in a combination of the Planck maps, the WMAP map, and the 408-MHz survey. The samples
of this exploration are used to infer the foreground marginalized low-ℓ likelihood
for any *TT* CMB spectrum.

** Inputs: **

- 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 of Haslam et al. (1982);
- Commander χ
^{2}-based LM93 confidence mask (Planck-2015-A10^{[5]}).

**File name and usage**
The commander likelihood is distributed in
*plc_2.0/low_l/commander/commander_rc2_v1.1_l2_29_B.clik*

When used with the library, this expects a vector of parameters consisting of the
*TT* CMB power spectrum from ℓ=0 to 29 (inclusive) and an extra nuisance parameter
consisting of the overall Planck calibration. Note that the vector really starts at ℓ=0, although
the first two entries are null.

**TEB **
This file allows for the computation of the CMB joint TT,EE, BB and TE likelihood
in the range ℓ=2-29. It should not be used with the low-ℓ TT-only likelihood.

** Production process**
The file allows for the computation of the pixel based likelihood of *T*, *E*, and *B*
maps. The *T* map is the best-fit map obtained from the commander algorithm, as described above,
while the *E* and *B* maps are obtained from the 70GHz LFI full mission data, but excluding the second and fourth surveys. Foreground contamination is
dealt with by marginalizing over templates based on the 30-GHz LFI and 353-GHz HFI maps.
The covariance matrix comes from the Planck detector sensitivity modulated on the
sky by the scanning strategy. The covariance matrix is further enlarged to account
for the foreground template removal.
To speed up the computation by about an order of magnitude, the problem is
projected into *C*_{ℓ} space using the Sherman-Morrison-Woodbury identity. Only the
projected quantities are stored in the data package.

** Inputs: **

- 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 of Haslam et al. (1982);
- Commander χ
^{2}-based LM93 confidence mask Planck-2015-A10^{[5]}.

**File name and usage**
The low ell TEB likelihood is distributed in
*plc_2.0/low_l/bflike/lowl_SMW_70_dx11d_2014_10_03_v5c_Ap.clik*.

When used with the library, this expects a vector of parameters consisting of the
*TT* CMB power spectrum from ℓ=0 to 29 (inclusive), followed by the *EE* (ℓ=0 to 29),
*BB* (ℓ=0 to 29) and *TE* (ℓ=0 to 29) spectra, and by an extra nuisance parameter
consisting of the overall Planck calibration. Note that the entries really start at ℓ=0, although the
ℓ=0 and ℓ=1 values will be null.

** High-ℓ likelihoods **

**TT only - Plik**
This file allows for the computation of the CMB TT likelihood in the range ℓ=30-2508.
Using the optional tool in the code package it can be modified to cover any multipole range within 29<ℓ<2509.

** Production process**
The file contains the 100-GHz, 143-GHz, and 217-GHz binned half-mission *TT* cross-spectra.
Only the 100×100, 143×143, 143×217, and 217×217 spectra are actually used. Masks and multipole
ranges for each spectrum are different and described in Planck-2015-A11^{[6]}.
Masks are based on the CMB-cleaned 353-GHz map for the dust component, on the Planck catalogues for the point source part, and on the CO maps.
The file also contains
templates for the residual foreground contamination of each spectrum. The templates are needed
to allow for computation of the joint CMB and nuisance likelihood. The covariance matrix is
computed using an analytical approximation, and corrected for the effect of point sources through Monte Carlo estimates.
The covariance matrix is computed for a fiducial
cosmology and nuisance model that has been obtained using a first, less optimal estimate of the parameters.

** Inputs: **

- Planck 100-, 143-, and 217-GHz half-mission
*T*maps; - CMB-cleaned (using the SMICA map) 353-GHz map, CO emission maps, and Planck catalogues for the masks;
- Planck 545-GHz maps for the dust residual contamination template;
- CIB, tSZ, kSZ, and CIB×SZ templates.

**File name and usage**
The high-ℓ, Plik TT likelihood is distributed in
*plc_2.0/hi_l/plik/plik_dx11dr2_HM_v18_TT.clik*.

When used with the library, this expects a vector of parameters consisting of the
*TT* CMB power spectrum from ℓ=0 to 2508 (inclusive), followed by a vector of 16 nuisance parameters.
Those are, in order:

- A_cib_217, the CIB contamination at ℓ=3000 in the 217-GHz Planck map;
- cib_index, the effective slope of the CIB spectrum, a parameter that should be set to -1.3;
- xi_sz_cib, the SZ×CIB cross-correlation;
- A_sz, the tSZ contamination at 143GHz;
- ps_A_100_100, the point source contribution in 100×100;
- ps_A_143_143, the point source contribution in 143×143;
- ps_A_143_217, the point source contribution in 143×217;
- ps_A_217_217, the point source contribution in 217×217;
- ksz_norm, the kSZ contamination;
- gal545_A_100, the dust residual contamination at ℓ=200 in 100×100;
- gal545_A_143, the dust residual contamination at ℓ=200 in 143×143;
- gal545_A_143_217, the dust residual contamination at ℓ=200 in 143×217;
- gal545_A_217, the dust residual contamination at ℓ=200 in 217×217;
- calib_100T, the relative calibration between the 100 and 143 spectra;
- calib_217T, the relative calibration between the 217 and 143 spectra;
- A_planck, the Planck absolute calibration.

We recommend using the following Gaussian priors:

- ksz_norm + 1.6 × A_sz = 9.5±3.0;
- cib_index = -1.3;
- gal545_A_100 = 7±2;
- gal545_A_143 = 9±2;
- gal545_A_143_217 = 21.0±8.5;
- gal545_A_217 = 80±20;
- calib_100T = 0.999±0.001;
- calib_217T = 0.995±0.002;
- A_planck = 1.0000±0.0025.

**TT+TE+EE - Plik**
This file allows for the computation of the CMB joint TT, TE, and EE likelihood in the range ℓ=30-2508 for TT and ℓ=30-1996 for TE and EE.

** Production process**
The file contains the 100-GHz, 143-GHz, and 217-GHz binned half-mission *T* and *E* cross-spectra.
In temperature, only the 100×100, 143×143, 143×217, and 217×217 spectra are actually used, while
in *TE* and *EE* all of them are used. Masks and multipole
ranges for each spectrum are different and described in Planck-2015-A11^{[6]}.
Masks are based on CMB-cleaned 353-GHz map for the dust component, on the Planck catalogues for the point source part, and on the CO maps.
The file also contains
templates for the residual foreground contamination of each spectrum. The templates are needed
to allow the computation of the joint CMB and nuisance likelihood. The covariance matrix is
computed using an analytical approximation, and corrected for the effect of point sources through Monte Carlo estimates.
The covariance matrix is computed for a fiducial
cosmology and nuisance model that has been obtained using a first, less optimal estimate of the parameters.

** Inputs: **

- Planck 100-, 143-, and 217-GHz half-mission T+P maps;
- CMB-cleaned (using the SMICA map) 353-GHz map, CO emission maps, and Planck catalogues for the masks;
- Planck 545-GHz and 353-GHz maps for the dust residual contamination template;
- CIB, tSZ, kSZ, and CIB×SZ templates;
- best-fit CMB+foreground model for the beam-leakage template.

**File name and usage**
The high-ℓ, plik TTTEEE likelihood is distributed in
*plc_2.0/hi_l/plik/plik_dx11dr2_HM_v18_TTTEEE.clik*.

This file should not be used with any other TT-only high-ℓ file.

When used with the library, this expects a vector of parameters consisting of the
*TT* CMB power spectrum from ℓ=0 to 2508 (inclusive), followed the *EE* and *TE* spectra (same range) and by a vector of 94 nuisance parameters.
Those are, in g:

- A_cib_217, the CIB contamination at ℓ=3000 in the 217-GHz Planck map;
- cib_index, the effective slope of the CIB spectrum, which should be set to -1.3;
- xi_sz_cib, the SZ×CIB cross-correlation;
- A_sz, the tSZ contamination at 143GHz;
- ps_A_100_100, the point source contribution in 100×100;
- ps_A_143_143, the point source contribution in 143×143;
- ps_A_143_217, the point source contribution in 143×217;
- ps_A_217_217, the point source contribution in 217×217;
- ksz_norm, the kSZ contamination;
- gal545_A_100, the dust residual contamination at ℓ=200 in 100×100TT;
- gal545_A_143, the dust residual contamination at ℓ=200 in 143×143TT;
- gal545_A_143_217, the dust residual contamination at ℓ=200 in 143×217TT;
- gal545_A_217, the dust residual contamination at ℓ=200 in 217×217TT;
- galf_EE_A_100, the dust residual contamination at ℓ=500 in 100×100EE;
- galf_EE_A_100_143, the dust residual contamination at ℓ=500 in 100×143EE;
- galf_EE_A_100_217, the dust residual contamination at ℓ=500 in 100×217EE;
- galf_EE_A_143, the dust residual contamination at ℓ=500 in 143×143EE;
- galf_EE_A_143_217, the dust residual contamination at ℓ=500 in 143×217EE;
- galf_EE_A_217, the dust residual contamination at ℓ=500 in 217×217EE;
- galf_EE_index, the dust EE template slope, which should be set to -2.4;
- galf_TE_A_100, the dust residual contamination at ℓ=500 in 100×100TE;
- galf_TE_A_100_143, the dust residual contamination at ℓ=500 in 100×143TE;
- galf_TE_A_100_217, the dust residual contamination at ℓ=500 in 100×217TE;
- galf_TE_A_143, the dust residual contamination at ℓ=500 in 143x143TE;
- galf_TE_A_143_217, the dust residual contamination at ℓ=500 in 143×217TE;
- galf_TE_A_217, the dust residual contamination at ℓ=500 in 217×217TE;
- galf_TE_index, the dust EE template slope, which should be set to -2.4;
- bleak_epsilon_0_0T_0E, beam-leakage parameter, espilon_0, 100×100 TE;
- bleak_epsilon_1_0T_0E, beam-leakage parameter, espilon_1, 100×100 TE;
- bleak_epsilon_2_0T_0E, beam-leakage parameter, espilon_2, 100×100 TE;
- bleak_epsilon_3_0T_0E, beam-leakage parameter, espilon_3, 100×100 TE;
- bleak_epsilon_4_0T_0E, beam-leakage parameter, espilon_4, 100×100 TE;
- bleak_epsilon_0_0T_1E, beam-leakage parameter, espilon_0, 100×143 TE;
- bleak_epsilon_1_0T_1E, beam-leakage parameter, espilon_1, 100×143 TE;
- bleak_epsilon_2_0T_1E, beam-leakage parameter, espilon_2, 100×143 TE;
- bleak_epsilon_3_0T_1E, beam-leakage parameter, espilon_3, 100×143 TE;
- bleak_epsilon_4_0T_1E, beam-leakage parameter, espilon_4, 100×143 TE;
- bleak_epsilon_0_0T_2E, beam-leakage parameter, espilon_0, 100×217 TE;
- bleak_epsilon_1_0T_2E, beam-leakage parameter, espilon_1, 100×217 TE;
- bleak_epsilon_2_0T_2E, beam-leakage parameter, espilon_2, 100×217 TE;
- bleak_epsilon_3_0T_2E, beam-leakage parameter, espilon_3, 100×217 TE;
- bleak_epsilon_4_0T_2E, beam-leakage parameter, espilon_4, 100×217 TE;
- bleak_epsilon_0_1T_1E, beam-leakage parameter, espilon_0, 143×143 TE;
- bleak_epsilon_1_1T_1E, beam-leakage parameter, espilon_1, 143×143 TE;
- bleak_epsilon_2_1T_1E, beam-leakage parameter, espilon_2, 143×143 TE;
- bleak_epsilon_3_1T_1E, beam-leakage parameter, espilon_3, 143×143 TE;
- bleak_epsilon_4_1T_1E, beam-leakage parameter, espilon_4, 143×143 TE;
- bleak_epsilon_0_1T_2E, beam-leakage parameter, espilon_0, 143×217 TE;
- bleak_epsilon_1_1T_2E, beam-leakage parameter, espilon_1, 143×217 TE;
- bleak_epsilon_2_1T_2E, beam-leakage parameter, espilon_2, 143×217 TE;
- bleak_epsilon_3_1T_2E, beam-leakage parameter, espilon_3, 143×217 TE;
- bleak_epsilon_4_1T_2E, beam-leakage parameter, espilon_4, 143×217 TE;
- bleak_epsilon_0_2T_2E, beam-leakage parameter, espilon_0, 217×217 TE;
- bleak_epsilon_1_2T_2E, beam-leakage parameter, espilon_1, 217×217 TE;
- bleak_epsilon_2_2T_2E, beam-leakage parameter, espilon_2, 217×217 TE;
- bleak_epsilon_3_2T_2E, beam-leakage parameter, espilon_3, 217×217 TE;
- bleak_epsilon_4_2T_2E, beam-leakage parameter, espilon_4, 217×217 TE;
- bleak_epsilon_0_0E_0E, beam-leakage parameter, espilon_0, 100×100 EE;
- bleak_epsilon_1_0E_0E, beam-leakage parameter, espilon_1, 100×100 EE;
- bleak_epsilon_2_0E_0E, beam-leakage parameter, espilon_2, 100×100 EE;
- bleak_epsilon_3_0E_0E, beam-leakage parameter, espilon_3, 100×100 EE;
- bleak_epsilon_4_0E_0E, beam-leakage parameter, espilon_4, 100×100 EE;
- bleak_epsilon_0_0E_1E, beam-leakage parameter, espilon_0, 100×143 EE;
- bleak_epsilon_1_0E_1E, beam-leakage parameter, espilon_1, 100×143 EE;
- bleak_epsilon_2_0E_1E, beam-leakage parameter, espilon_2, 100×143 EE;
- bleak_epsilon_3_0E_1E, beam-leakage parameter, espilon_3, 100×143 EE;
- bleak_epsilon_4_0E_1E, beam-leakage parameter, espilon_4, 100×143 EE;
- bleak_epsilon_0_0E_2E, beam-leakage parameter, espilon_0, 100×217 EE;
- bleak_epsilon_1_0E_2E, beam-leakage parameter, espilon_1, 100×217 EE;
- bleak_epsilon_2_0E_2E, beam-leakage parameter, espilon_2, 100×217 EE;
- bleak_epsilon_3_0E_2E, beam-leakage parameter, espilon_3, 100×217 EE;
- bleak_epsilon_4_0E_2E, beam-leakage parameter, espilon_4, 100×217 EE;
- bleak_epsilon_0_1E_1E, beam-leakage parameter, espilon_0, 143×143 EE;
- bleak_epsilon_1_1E_1E, beam-leakage parameter, espilon_1, 143×143 EE;
- bleak_epsilon_2_1E_1E, beam-leakage parameter, espilon_2, 143×143 EE;
- bleak_epsilon_3_1E_1E, beam-leakage parameter, espilon_3, 143×143 EE;
- bleak_epsilon_4_1E_1E, beam-leakage parameter, espilon_4, 143×143 EE;
- bleak_epsilon_0_1E_2E, beam-leakage parameter, espilon_0, 143×217 EE;
- bleak_epsilon_1_1E_2E, beam-leakage parameter, espilon_1, 143×217 EE;
- bleak_epsilon_2_1E_2E, beam-leakage parameter, espilon_2, 143×217 EE;
- bleak_epsilon_3_1E_2E, beam-leakage parameter, espilon_3, 143×217 EE;
- bleak_epsilon_4_1E_2E, beam-leakage parameter, espilon_4, 143×217 EE;
- bleak_epsilon_0_2E_2E, beam-leakage parameter, espilon_0, 217×217 EE;
- bleak_epsilon_1_2E_2E, beam-leakage parameter, espilon_1, 217×217 EE;
- bleak_epsilon_2_2E_2E, beam-leakage parameter, espilon_2, 217×217 EE;
- bleak_epsilon_3_2E_2E, beam-leakage parameter, espilon_3, 217×217 EE;
- bleak_epsilon_4_2E_2E, beam-leakage parameter, espilon_4, 217×217 EE;
- calib_100T, the relative calibration between 100 and 143 TT spectra;
- calib_217T, the relative calibration between 217 and 143 TT spectra;
- calib_100P, the calibration of the 100 EE spectra, which should be set to 1;
- calib_143P, the calibration of the 143 EE spectra, which should be set to 1;
- calib_217P, the calibration of the 217 EE spectra, which should be set to 1;
- A_pol, the calibration of the polarization relative to the temperature, which should be set to 1;
- A_planck, the Planck absolute calibration.

We recommend using the following Gaussian priors:

- ksz_norm + 1.6 × A_sz = 9.5±3.0;
- cib_index = -1.3;
- gal545_A_100 = 7±2;
- gal545_A_143 = 9±2;
- gal545_A_143_217 = 21.0±8.5;
- gal545_A_217 = 80±20;
- galf_EE_A_100 = 0.060±0.012;
- galf_EE_A_100_143 = 0.050±0.015;
- galf_EE_A_100_217 = 0.110±0.033;
- galf_EE_A_143 = 0.10±0.02;
- galf_EE_A_143_217 = 0.240±0.048;
- galf_EE_A_217 = 0.72±0.14;
- galf_EE_index = -2.4;
- galf_TE_A_100 = 0.140±0.042;
- galf_TE_A_100_143 = 0.120±0.036;
- galf_TE_A_100_217 = 0.30±0.09;
- galf_TE_A_143 = 0.240±0.072;
- galf_TE_A_143_217 = 0.60±0.018;
- galf_TE_A_217 = 1.80±0.54;
- galf_TE_index = -2.4;
- bleak_epsilon_0_0T_0E = 0;
- bleak_epsilon_1_0T_0E = 0;
- bleak_epsilon_2_0T_0E = 0;
- bleak_epsilon_3_0T_0E = 0;
- bleak_epsilon_4_0T_0E = 0;
- bleak_epsilon_0_0T_1E = 0;
- bleak_epsilon_1_0T_1E = 0;
- bleak_epsilon_2_0T_1E = 0;
- bleak_epsilon_3_0T_1E = 0;
- bleak_epsilon_4_0T_1E = 0;
- bleak_epsilon_0_0T_2E = 0;
- bleak_epsilon_1_0T_2E = 0;
- bleak_epsilon_2_0T_2E = 0;
- bleak_epsilon_3_0T_2E = 0;
- bleak_epsilon_4_0T_2E = 0;
- bleak_epsilon_0_1T_1E = 0;
- bleak_epsilon_1_1T_1E = 0;
- bleak_epsilon_2_1T_1E = 0;
- bleak_epsilon_3_1T_1E = 0;
- bleak_epsilon_4_1T_1E = 0;
- bleak_epsilon_0_1T_2E = 0;
- bleak_epsilon_1_1T_2E = 0;
- bleak_epsilon_2_1T_2E = 0;
- bleak_epsilon_3_1T_2E = 0;
- bleak_epsilon_4_1T_2E = 0;
- bleak_epsilon_0_2T_2E = 0;
- bleak_epsilon_1_2T_2E = 0;
- bleak_epsilon_2_2T_2E = 0;
- bleak_epsilon_3_2T_2E = 0;
- bleak_epsilon_4_2T_2E = 0;
- bleak_epsilon_0_0E_0E = 0;
- bleak_epsilon_1_0E_0E = 0;
- bleak_epsilon_2_0E_0E = 0;
- bleak_epsilon_3_0E_0E = 0;
- bleak_epsilon_4_0E_0E = 0;
- bleak_epsilon_0_0E_1E = 0;
- bleak_epsilon_1_0E_1E = 0;
- bleak_epsilon_2_0E_1E = 0;
- bleak_epsilon_3_0E_1E = 0;
- bleak_epsilon_4_0E_1E = 0;
- bleak_epsilon_0_0E_2E = 0;
- bleak_epsilon_1_0E_2E = 0;
- bleak_epsilon_2_0E_2E = 0;
- bleak_epsilon_3_0E_2E = 0;
- bleak_epsilon_4_0E_2E = 0;
- bleak_epsilon_0_1E_1E = 0;
- bleak_epsilon_1_1E_1E = 0;
- bleak_epsilon_2_1E_1E = 0;
- bleak_epsilon_3_1E_1E = 0;
- bleak_epsilon_4_1E_1E = 0;
- bleak_epsilon_0_1E_2E = 0;
- bleak_epsilon_1_1E_2E = 0;
- bleak_epsilon_2_1E_2E = 0;
- bleak_epsilon_3_1E_2E = 0;
- bleak_epsilon_4_1E_2E = 0;
- bleak_epsilon_0_2E_2E = 0;
- bleak_epsilon_1_2E_2E = 0;
- bleak_epsilon_2_2E_2E = 0;
- bleak_epsilon_3_2E_2E = 0;
- bleak_epsilon_4_2E_2E = 0;
- calib_100T = 0.999±0.001;
- calib_217T = 0.995±0.002;
- calib_100P = 1;
- calib_143P = 1;
- calib_217P = 1;
- A_pol = 1;
- A_planck = 1.0000±0.0025.

**TT only - Plik lite**
This file allows for the computation of the nuisance-marginalized CMB TT likelihood in the range ℓ=30-2508. It should not be used with the regular high-ℓ TT or TTTEEE files.

** Production process**
The Plik likelihood files described above have been explored using a Bayesian algorithm described in Planck-2015-A11^{[6]}. The joint posterior of the CMB *TT* spectrum, marginalized over the nuisance parameters, has been extracted from this analysis to build a high-ℓ likelihood approximation for the
CMB spectrum only. The nuisance parameters have been marginalized in the priors described above.

** Inputs: **

- Plik plik_dx11dr2_HM_v18_TT likelihood;
- dust residual, CIB, tSZ, kSZ, and CIB×SZ templates.

**File name and usage**
The high-ℓ, Plik TT, nuisance-marginalized likelihood is distributed in
*plc_2.0/hi_l/plik_lite/plik_lite_v18_TT.clik*.

When used with the library, this expects a vector of parameters consisting of the
*TT* CMB power spectrum from ℓ=0 to 2508 (inclusive), followed by the Planck absolute calibration nuisance parameter.

**TT EE TE - Plik lite**
This file allows for the computation of the nuisance marginalized CMB joint TT, TE, EE likelihood in the range ℓ=30-2508 for temperature and ℓ=30-1996 for TE and EE. It should not be used with the regular high-ℓ TT or TTTEEE files.

** Production process**
The Plik likelihood file described above have been explored using a Bayesian algorithm described in Planck-2015-A11^{[6]}. The joint posterior of the CMB *TT, TE, and EE spectra, marginalized over the nuisance parameters has been extracted from this analysis to build a high-ℓ likelihood approximation for the
CMB spectrum only. The nuisance parameters have been marginalized in the priors described above.
*

** Inputs: **

- Plik plik_dx11dr2_HM_v18_TTTEEE likelihood;
- dust residual, CIB, tSZ, kSZ, and CIB×SZ templates.

**File name and usage**
The high-ℓ, Plik TTTEEE, nuisance-marginalized likelihood is distributed in
*plc_2.0/hi_l/plik_lite/plik_lite_v18_TTTEEE.clik*.

When used with the library, this expects a vector of parameters consisting of the
*TT* CMB power spectrum from ℓ=0 to 2508 (inclusive), followed by the *EE*, and *TE* spectra (same range) and by the Planck absolute calibration nuisance parameter.

** Lensing likelihoods **

**T only**
This file allows for the computation of the baseline lensing likelihood, using only the SMICA *T*
map-based lensing reconstruction. Only the lensing multipole range ℓ=40-400
is used. The covariance matrix for the likelihood is based on Monte Carlos. The likelihood includes
the computation of a model-dependent correction to the normalization and
the "N1 bias." To speed up those computations, both are performed using a linear approximation.

** Production process**
The SMICA *T* maps are filtered and correlated to reconstruct an optimal
lensing reconstruction map. Biases are corrected using Monte Carlo simulations
for the dominant "mean field" and "N0" contribution, while the "N1" bias is computed using the CMB-based best-fit model. The power spectrum of this map is measured out of a large mask, and binned.
The covariance matrix of this binned spectrum is evaluated using numerical simulations.
The model-dependent corrections to the normalization and N1 biases are linearized and precomputed.

** Inputs: **

- SMICA
*T*CMB map; - 60% Galactic mask;
- best-fit Planck CMB model.

**File name and usage**
The T-only lensing likelihood is distributed in
*plc_2.0/hi_l/lensing/smica_g30_ftl_full_pttptt.clik_lensing*.

When used with the library, this expects a vector of parameters consisting of the
φφ lensing spectrum for ℓ=0 to 2048 (inclusive), followed by the
*TT* CMB power spectrum for ℓ=0 to 2048 (inclusive) and by the Planck absolute calibration nuisance parameter.

The *TT* spectrum is needed to compute the normalization and N1 corrections.

**T+P**
This file allows for the computation of the baseline lensing likelihood, using both the
*T* and *P* SMICA map-based lensing reconstruction. Only the lensing multipole rangeℓ=40-400
is used. The covariance matrix for the likelihood is based on Monte Carlos. The likelihood includes
the computation of model dependent correction to the normalization and
the N1 bias. To speed up those computations, both are performed using a linear approximation.

** Production process**
The SMICA *T* and *P* maps are filtered and correlated to reconstruct an optimal
lensing reconstruction map. Biases are corrected using Monte Carlo simulations
for the dominant mean field and N0 contribution, while the N1 bias is computed using the CMB-based best-fit model. The power spectrum of this map is measured outside a large mask, and binned.
The covariance matrix of this binned spectrum is evaluated using numerical simulations.
The model-dependent corrections to the normalization and N1 biases are linearized and precomputed.

** Inputs: **

- SMICA
*T*and*E*CMB maps; - 60% Galactic mask;
- best-fit Planck CMB model.

**File name and usage**
The T+P-only lensing likelihood is distributed in
*plc_2.0/hi_l/lensing/smica_g30_ftl_full_pp.clik_lensing*.

When used with the library, this expects a vector of parameters consisting of the
φφ lensing spectrum for ℓ=0 to 2048 (inclusive), followed by the
*TT* CMB power spectrum for ℓ=0 to 2048 (inclusive), the *EE* and *TE* power spectra (same range) and by the Planck absolute calibration nuisance parameter.

The *TT*, *EE*, and *TE* spectra are needed to compute the normalization and N1 corrections.

**Data sets - Extended data**

Four other data files are available. Those extend the baseline delivery by adding:

- TE, EE and other joint high-ℓ Plik likelihoods,
*COM_Likelihood_Data-extra-plik-HM-ext_R2.00.tar.gz*; - TT and TTTEEE unbinned high-ℓ Plik likelihood,
*COM_Likelihood_Data-extra-plik-unbinned_R2.00.tar.gz*; - TT, TE, EE, and TTTEEE Plik likelihood using the detset cross-spectra instead of the half-mission one, and including empirical correlated noise correction for the TT spectra,
*COM_Likelihood_Data-extra-plik-DS_R2.00.tar.gz*; - extended ℓ range lensing likelihood,
*COM_Likelihood_Data-extra-lensing-ext_R2.00.tar.gz*.

Note that although these products are discussed in the Planck papers, they are not used for the baseline results.

Each file contains a **readme.md** description of the content and the use of the different likelihood files.

**Likelihood Masks**

**General description**

We provide the masks for Temperature and Polarisation used to exclude the regions of strongest foreground emission in the CMB maps when computing the empirical spectra that we use to form the high-ℓ likelihood. The masks are provided for each frequency between 100 and 217 GHz, masking the appropriate part of the Galactic Plane and, for temperature only, the relevant population of point sources. Note that to compute the empirical power spectrum of the maps, one should also take into account missing pixels in a particular map. This is not included in the masks we distribute.

**Production Process**
Complete production process for those masks is described in Planck-2015-A11^{[6]}.

The temperature masks are the combination of a Galactic mask, a point source mask and (for 100Ghz and 217GHz) a CO mask. Extended objects, planets, etc are included in the point source mask. The polarization masks are simply the Galactic masks.

The Galactic masks are obtained by thresholding a CMB corrected and smoothed 353GHz map. The threshold is tuned so as to retain a given fraction of the sky, between 50% and 80%. Each mask is apodized by a FWHM gaussian window function using the distance map. Special care is taken to smooth this distance map, in order to avoid strong caustics. The apodization down-weights an effective 10% of the sky. Following Planck-2015-A11^{[6]}, the Galactic masks are named by the effective unmasked sky area G70, G60, G50 and G41.

The point source masks are different for each frequency. They are based on the 2015 Planck catalog and cut out all the sources detected at S/N = 5 or higher in each channel. Each source is masked with a circular aperture of size 3xFWHM of that channel (i.e. 3x9.′66 at 100GHz, 3x7.′22 at 143GHz, and 3x4.′90 at 217GHz). Note that we include in the point source masks all sources above our threshold, including possible galactic ones. The mask is further apodized with a gaussian window function with FWHM=30'.

The CO masks is used only at 100GHz and 217GHz, since there is no (strong) CO line in the 147 GHz channel. They are based on the Type 3 CO map from the Planck2013 delivery. The CO map is smoothed to 120' and a threshold is applied to mask all regions above

. The mask is further apodized to 30'.Mask combinations are performed by multiplying a galactic, a point source, and possibly a CO mask together.

The Temperature masks are named in Planck-2015-A11^{[6]} using the capital letter T and the effective sky fraction that is left unmasked. They are produced using the following

- 100Ghz: T66 = G70+Point source mask 100Ghz+CO mask
- 143Ghz: T57 = G60+Point source mask 143Ghz
- 217Ghz: T47 = G50+Point source mask 217Ghz+CO mask

The Polarization masks are named in Planck-2015-A11^{[6]} using the capital letter P and the effective sky fraction that is left unmasked. They are produced using the following

- 100Ghz: P70 = G70
- 143Ghz: P50 = G50
- 217Ghz: P41 = G41

**Inputs**

- Galactic masks: The SMICA CMB temperature map, the 353GHz temperature map.
- Point source masks: The Planck 2015 catalog, the FWHM of each of the frequencies.
- CO mask: the Type 3 CO map from the Planck 2013 delivery.

** File names and meta-data **
The masks are distributed in a single tar file *COM_Likelihood_Masks_R2.00.tar.gz*.
The files extract to 6 files

- temperature_100.fits: an nside=2048 healpix map of double precision, containing the Temperature mask used at 100Ghz, i.e. T66
- temperature_143.fits: an nside=2048 healpix map of double precision, containing the Temperature mask used at 143Ghz, i.e. T57
- temperature_217.fits: an nside=2048 healpix map of double precision, containing the Temperature mask used at 217Ghz, i.e. T47
- polarization_100.fits: an nside=2048 healpix map of double precision, containing the Polarization mask used at 100Ghz, i.e. P70
- polarization_143.fits: an nside=2048 healpix map of double precision, containing the Polarization mask used at 143Ghz, i.e. P50
- polarization_217.fits: an nside=2048 healpix map of double precision, containing the Polarization mask used at 217Ghz, i.e. P41

*
*

**2013 CMB spectrum and Likelihood**

**General description**

**CMB spectra**

The Planck best-fit CMB temperature power spectrum, shown in figure below, covers the wide range of multipoles = 2-2479. Over the multipole range = 2–49, 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 Planck-2013-XII^{[12]}. The asymmetric error bars associated to this spectrum are the 68% confidence limits and include the uncertainties due to foreground subtraction. For multipoles greater than , instead, the spectrum is derived from the *CAMspec* likelihood Planck-2013-XV^{[8]} by optimally combining the spectra in the frequency range 100-217 GHz, and correcting them for unresolved foregrounds. Associated 1-sigma errors include beam and foreground uncertainties. Both *Commander* and *CAMspec* are described in more details in the sections below.

**Likelihood**

The likelihood code (and the data that comes with it) used to compute the likelihood of a model that predicts the CMB power spectra, lensing power spectrum, together with some foreground and some instrumental parameters. The data files are built primarily from the Planck mission results, but include also some results from the WMAP-9 data release. The data files are written in a specific format that can only be read by the code. The code consists in a c/f90 library, along with some optional tools in python. The code is used to read the data files, and given model power spectra and nuisance parameters it computes the log likelihood of that model.

Detailed description of the installation and usage of the likelihood code and data is provided in the package. The package includes five data files: four for the CMB likelihoods and one for the lensing likelihood. All of the likelihoods delivered are described in detail in the Power spectrum & Likelihood Paper Planck-2013-XV^{[8]} (for the CMB based likelihood) and in the Lensing Paper (for the lensing likelihood) Planck-2013-XVII^{[14]}.

The CMB full likelihood has been divided into four parts to allow using selectively different ranges of multipoles. It also reflects the fact that the mathematical approximations used for those different parts are very different, as is the underlying data. In detail, we distribute

- one low- temperature only likelihood (commander),
- one low- temperature and polarisation likelihood (lowlike), and
- one higl- likelihood CAMspec.

The *Commander* likelihood covers the multipoles 2 to 49. It uses a semi-analytic method to sample the low- temperature likelihood on an intermediate product of one of the component separated maps. The samples are used along with an analytical approximation of the likelihood posterior to perform the likelihood computation in the code. See Planck-2013-XV^{[8]} section 8.1 for more details.

The *lowlike* likelihood covers the multipoles 2 to 32 for temperature and polarization data. Since Planck is not releasing polarisation data at this time, the polarization map from WMAP9 is used instead. A temperature map is needed to perform the computation nevertheless, and we use here the same commander map. The likelihood is computed using a map-based approximation at low resolution and a master one at intermediate resolution, as in WMAP. The likelihood code actually calls a very slightly modified version of the WMAP9 code. This piece of the likelihood essentially provides a prior on the optical depth and has almost no other impact on cosmological parameter estimation. As such it could be replaced by a simple prior, and a user can decide to do so, which is one of the motivation to leave the three pieces of the CMB likelihood as different data packages; see Planck-2013-XV^{[8]} section 8.3 for more details. Note that the version of the WMAP code used here (code version v1.0) does not perform any test on the positive definiteness of the TT, TE, EE covariance matrices, and will return a null log likelihood in the unphysical cases where the absolute value of TE is too large. This will be corrected in a later version.

The *CAMspec* likelihood covers the multipoles 50 to 2500 for temperature only. The likelihood is computed using a quadratic approximation, including mode to mode correlations that have been precomputed on a fiducial model. The likelihood uses data from the 100, 143 and 217 GHz channels. To do so it models the foreground at each frequency using the model described in the likelihood paper. Uncertainties on the relative calibration and on the beam transfer functions are included either as parametric models, or marginalized and integrated in the covariance matrix. Detailed description of the different nuisance parameters is given below. Priors are included in the likelihood on the CIB spectral index, relative calibration factors and beam error eigenmodes. See Planck-2013-XV^{[8]} section 2.1 for more details.

The *act/spt* likelihood covers the multipoles 1500 to 10000 for temperature. It is described in^{[15]}^{[16]}^{[17]}. It uses the code and data that can be retrieved from the Lambda archive for ACT and SPT. It has been slightly modified to use a thermal and kinetic SZ model that matches the one used in CAMspec. As stated in^{[15]}, the dust parameters a_ge and a_gs must be explored with the following priors: a_ge = 0.8 ± 0.2 and a_gs = 0.4 ± 0.2. Those priors are not included in the log likelihood computed by the code.

The *lensing* likelihood covers the multipoles 40 to 400 using the result of the lensing reconstruction. It uses a quadratic approximation for the likelihood, with a covariance matrix including the marginalized contribution of the beam transfer function uncertainties, the diffuse point source correction uncertainties and the cosmological model uncertainty affecting the first order non-gaussian bias (N1). The correlation between temperature and lensing is not taken into account. Cosmological uncertainty effects on the normalization are dealt with using a first order renormalization procedure. This means that the code will need both the TT and $\phi\phi$ power spectrum up to = 2048 to correctly perform the integrals needed for the renormalization. Nevertheless, the code will only produce an estimate based on the data between = 40 to 400. See Planck-2013-XVII^{[14]} section 6.1 for more details.

**Production process**

**CMB spectra**

The Planck-2013-XII^{[12]}. 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^{[8]}.

The CMB temperature power spectrum has been derived by the CamSpec likelihood, a code that implements a pseudo-Cl based technique, extensively described in Sec. 2 and the Appendix of Planck-2013-XV^{[8]}. Frequency spectra are computed as noise weighted averages of the cross-spectra between single detector and sets of detector maps. Mask and multipole range choices for each frequency spectrum are summarized in Table 4 of Planck-2013-XV^{[8]}. 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+WP+highL in Table 5 of Planck-2013-XVI^{[13]}). A thorough description of the models of unresolved foregrounds is given in Sec. 3 of Planck-2013-XV^{[8]} and Sec. 4 of Planck-2013-XVI^{[13]}. The spectrum covariance matrix accounts for cosmic variance and noise contributions, together with unresolved foreground and beam uncertainties. Both spectrum and associated covariance matrix are given as uniformly weighted band averages in 74 bins.

**Likelihood**

The code is based on some basic routines from the libpmc library in the cosmoPMC code. It also uses some code from the WMAP9 likelihood for the lowlike likelihood and^{[15]}^{[16]}^{[17]} for the act/spt one. The rest of the code has been specifically written for the Planck data. Each likelihood file has been processed using a different and dedicated pipeline as described in the likelihood paper Planck-2013-XV^{[8]} (section 2 and 8) and in the lensing paper Planck-2013-XVII^{[14]} (section 6.1). We refer the reader to those papers for full details. The data are then encapsulated into the specific file format.

Each dataset comes with its own self check. Whenever the code is used to read a data file, a computation will be done against an included test spectrum/nuisance parameter, and the log-likelihood will be displayed along with the expected result. Difference of the order of 10

or less are expected depending of the architecture.**Inputs**

**CMB spectra**

- Low-l spectrum ( )

- frequency maps from 30–353 GHz
- common mask Planck-2013-XII
^{[12]} - compact sources catalog

- High-l spectrum ( )

- 100, 143, 143x217 and 217 GHz spectra and their covariance matrix (Sec. 2 in Planck-2013-XV
^{[8]}) - best-fit foreground templates and inter-frequency calibration factors (Table 5 of Planck-2013-XVI
^{[13]}) - Beam transfer function uncertainties Planck-2013-VII
^{[18]}

**Likelihood**

*commander*

- all Planck channels maps
- compact source catalogs
- common masks
- beam transfer functions for all channels

*lowlike*

- WMAP9 likelihood data
- Low- Commander map

*CAMspec*

- 100, 143 and 217 GHz detector and detsets maps
- 857GHz channel map
- compact source catalog
- common masks (0,1 & 3)
- beam transfer function and error eigenmodes and covariance for 100, 143 and 217 GHz detectors & detsets
- theoretical templates for the tSZ and kSZ contributions
- color corrections for the CIB emission for the 143 and 217GHz detectors and detsets
- fiducial CMB model (bootstrapped from WMAP7 best fit spectrum) estimated noise contribution from the half-ring maps for 100, 143 and 217 GHz

*lensing*

- the lensing map
- beam error eigenmodes and covariance for the 143 and 217GHz channel maps
- fiducial CMB model (from Planck cosmological parameter best fit)

*act/spt*

- data and code from here
- the tSZ andkSZ template are changed to match those of CAMspec

** File names and Meta data **

**CMB spectra**

The CMB spectrum and its covariance matrix are distributed in a single FITS file named

which contains 3 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): $D_l$ as described below*ERRUP*(float): the upward uncertainty*ERRDOWN*(float): the downward uncertainty

- HIGH-ELL (BINTABLE)
- with the high-ell part of the spectrum, binned into 74 bins covering in bins of width (with the exception of the last 4 bins that are wider). 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): $D_l$ as described below*ERR*(float): the uncertainty

- COV-MAT (IMAGE)
- with the covariance matrix of the high-ell part of the spectrum in a 74x74 pixel image, i.e., covering the same bins as the
*HIGH-ELL*table.

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 $\mu\, K^4$. The spectra are shown in the figure below, in blue and red for the low- and high-

parts, respectively, and with the error bars for the high-ell part only in order to avoid confusion.**Likelihood**

**Likelihood source code**

The source code is in the file

- COM_Code_Likelihood-v1.0_R1.10.tar.gz (C, f90 and python likelihood library and tools)

**Likelihood data packages**

The data packages are

*COM_Data_Likelihood-commander_R1.10.tar.gz*(low-ell TT likelihood)*COM_Data_Likelihood-lowlike_R1.10.tar.gz*(low-ell TE,EE,BB likelihood)*COM_Data_Likelihood-CAMspec_R1.10.tar.gz*(high-ell TT likelihood)*COM_Data_Likelihood-actspt_R1.10.tar.gz*(high-ell TT likelihood)*COM_Data_Likelihood-lensing_R1.10.tar.gz*(lensing likelihood)

Untar and unzip all files to recover the code and likelihood data. Each package comes with a README file; follow the instructions inclosed to build the code and use it. To compute the CMB likelihood one has to sum the log likelihood of each of the commander_v4.1_lm49.clik, lowlike_v222.clik and CAMspec_v6.2TN_2013_02_26.clik, actspt_2013_01.clik. To compute the CMB+lensing likelihood, one has to sum the log likelihood of all 5 files.

The CMB and lensing likelihood format are different. The CMB files have the termination .clik, the lensing one .clik_lensing. The lensing data being simpler (due to the less detailled modeling permitted by the lower signal-noise), the file is a simple ascii file containing all the data along with comments describing it, and linking the different quantities to the lensing paper. The CMB file format is more complex and must accommodate different forms of data (maps, power spectrum, distribution samples, covariance matrices...). It consists of a tree structure containing the data. At each level of the tree structure a given directory can contain array data (in the form of FITS files or ascii files for strings) and scalar data (joined in a single ascii file "_mdb"). Those files are not user modifiable and do not contain interesting meta data for the user. Tools to manipulate those files are included in the code package as optional python tools. They are documented in the code package.

**Likelihood masks**

The masks used in the Likelihood paper Planck-2013-XV^{[8]} are found in
COM_Mask_Likelihood_2048_R1.10.fits

which contains ten masks which are written into a single *BINTABLE* extension of 10 columns by 50331648 rows (the number of Healpix pixels in an Nside = 2048 map). The structure is as follows, where the column names are the names of the masks:

1. EXTNAME = 'MSK-LIKE' : Data columns | |||
---|---|---|---|

Column Name | Data Type | Units | Description |

CL31 | Real*4 | none | mask |

CL39 | Real*4 | none | mask |

CL49 | Real*4 | none | mask |

G22 | Real*4 | none | mask |

G35 | Real*4 | none | mask |

G45 | Real*4 | none | mask |

G56 | Real*4 | none | mask |

G65 | Real*4 | none | mask |

PS96 | Real*4 | none | mask |

PSA82 | Real*4 | none | mask |

Keyword | Data Type | Value | Description |

PIXTYPE | string | HEALPIX | |

COORDSYS | string | GALACTIC | Coordinate system |

ORDERING | string | NESTED | Healpix ordering |

NSIDE | Int | 2048 | Healpix Nside |

FIRSTPIX | Int*4 | 0 | First pixel number |

LASTPIX | Int*4 | 50331647 | Last pixel number, for LFI and HFI, respectively |

## References[edit]

- ↑
^{1.0}^{1.1}^{1.2}**Planck 2018 results. VI. Cosmological parameters**, Planck Collaboration, 2020, A&A, 641, A6. - ↑
^{2.0}^{2.1}^{2.2}**Planck 2018 results. V. CMB Power Spectra and Likelihoods**, Planck Collaboration, 2020, A&A, 641, A5. - ↑
**Planck 2018 results. IV. Diffuse component separation**, Planck Collaboration, 2020, A&A, 641, A4. - ↑
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^{6.00}^{6.01}^{6.02}^{6.03}^{6.04}^{6.05}^{6.06}^{6.07}^{6.08}^{6.09}^{6.10}^{6.11}^{6.12}^{6.13}^{6.14}^{6.15}^{6.16}^{6.17}**Planck 2015 results. XI. CMB power spectra, likelihoods, and robustness of cosmological parameters**, Planck Collaboration, 2016, A&A, 594, A11. - ↑
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**Planck 2013 results. VII. HFI time response and beams**, Planck Collaboration, 2014, A&A, 571, A7.

*
*

Cosmic Microwave background

(Planck) High Frequency Instrument

Flexible Image Transfer Specification

(Planck) Low Frequency Instrument

Sunyaev-Zel'dovich

Full-Width-at-Half-Maximum