Cosmological Parameters

Description

The 2018 cosmological parameter results explore a variety of cosmological models with combinations of Planck and other data. We provide results from MCMC exploration chains, as well as best fits, and sets of parameter tables. Definitions, conventions and reference are contained in Planck-2013-XVIPlanck-2015-A15 and 2018 parameter paper.

Production process

Parameter chains are produced using CosmoMC, a sampling package available here. This includes the sample analysis package (and GUI) GetDist, and the scripts for managing, analysing, and plotting results from the full grid or runs. The Python GetDist sample analysis package is also available separately.

Chain products provided here have had burn in removed. Some results with additional data are produced by importance sampling. Note that the baseline model includes one massive neutrino (0.06eV).

Caveats and known issues

1. Confidence intervals are derived from the MCMC samples, and assume the input likelihoods are exactly correct, so there is no quantification for systematic errors other than via the covariance, foreground and beam error models assumed in the likelihood codes.
2. Non-linear lensing modelling uses Halofit (HMCode); for some extended models and CMB lensing only analyses, tails of the chains may be away from the domain of validity.
3. Polarization results are sensitive to details of the polarization modelling; systematic uncertainties is not accounted for and may change results by up to one standard deviation in some cases but is difficult to fully quantify. Alternative CamSpecHM likelihood results are also provided and can give some idea of differences that can be obtained with different analysis choices.
4. Quoted values for the helium abundance (Y_P and Y_P^BBN) and D/H, when Y_P is not varied as a parameter, are obtained assuming the Parthenope 1.1 BBN code, and do not include theoretical errors.
5. Some nuisance parameters have slightly different definitions in the plikHM and CamSpecHM likelihoods, and cannot be directly compared.
6. Tables include DES likelihoods, but not chain files. These will be provided once the DES likelihoods are published
7. Bicep-Keck joint results are not included. These will be added when the latest release is available.

Related products

Results of the baseline parameter exploration runs should be reproducible using CosmoMC with the Planck 2018 likelihood codes. A new Python code called Cobaya also follows the same methodology and should produce identical results.

Parameter Tables

These list parameter constraints for each considered model and data combination separately. For the baseline likelihood combinations see

• PDF tables with 68% limits Media:baseline_params_table_2018_68pc.pdf
• PDF tables with 95% limits Media:baseline_params_table_2018_95pc.pdf

There are also larger full files including results from the alternative CamSpec likelihood, lensing only, other data combinations and priors:

• PDF tables with 68% limits Media:result_table_2018_68pc.pdf
• PDF tables with 95% limits Media:result_table_2018_95pc.pdf

A full set of parameter tables, including various ways of comparing data combinations, are available in one bundle from the PLA as shown below.

Data combination tags used to label results are as follows (see for full description and references):

Tag	Data
plikHM	baseline high-L Planck power spectra (plik cross half-mission, 30 <= l <= 2508)
CamSpecHM	high-L Planck power spectra (CamSpec cross half-mission, 30 <= l <= 2500)
CleanedCamSpecHM	foreground-cleaned high-L Planck CamSpec spectra (cross half-mission, 30 <= l <= 2500)
lowl	low-L: Planck temperature only (2 <= l <= 29)
lowE	low-L HFI polarization EE likelihood (2 <= l <= 29)
lensing	Planck lensing conservative power spectrum reconstruction likelihood
zre6p5	A hard prior z_re > 6.5
reion	A hard prior z_re > 6.5, combined with Gaussian prior z_re = 7 +- 1
BAO	Baryon oscillation data from DR12, MGS and 6DF
Pantheon	Supernova data from the Pantheon sample
JLA	Supernova data from the SDSS-II/SNLS3 Joint Light-curve Analysis
Riess18	Local Hubble parameter measurement from Riess et al.(a), H_0 = 73.45 +- 1.66
theta	theta_MC = 1.0409 +- 0.0006 Gaussian prior
WMAP	The full WMAP (temperature and polarization) 9 year data
lenspriors	Standard base parameters with n_s = 0.96 += 0.02, Omega_b h^2 = 0.0222 +- 0.0005, 100>H_0>40, tau=0.055.
DESpriors	DES cosmological parameter priors (flat on 0.1< Omega_m<0.9, 0.03<Omega_b<0.07, 55<H_0<91, 0.5<10^9A_s<5, Y_P=0.245341 and,if varied, 0.05 eV< sum m_nu <1 eV)
DES	DES 1yr, cosmic shear+galaxy auto+cross
DESlens	DES 1yr, cosmic shear only
DESwt	DES 1yr, galaxy auto+cross only

The high-L Planck likelihoods have TT, TE, EE variants from each spectrum alone, plus the TTTEEE joint constraint. When the lensing likelihood is used with DESpriors or lenspriors, it is marginalized over the theoretical CMB power spectra (as described in the 2018 lensing paper).

Tags used to identify the model parameters that are varied are described in the introduction to the PDF table files.

Parameter Chains

We provide the full chains and getdist outputs for our parameter results. The entire grid of results, including likelihood variations, various external data combinations and CMB lensing only results, is available s a 9GB compressed file:

• COM_CosmoParams_fullGrid_R3.00.zip

Where nn is the most recent update. You can also download smaller files containing key results:

• COM_CosmoParams_base-plikHM-TTTEEE-lowl-lowE_R3.00.zip
• COM_CosmoParams_base-plikHM_R3.00.zip
• COM_CosmoParams_base-plikHM-zre6p5_R3.00.zip
• COM_CosmoParams_lensonly_R3.00.zip
• COM_CosmoParams_parameter-tables_R3.00.zip

The download contains a hierarchy of directories, with each separate chain in a separate directory. The structure for the directories is

base_AAA_BBB/XXX_YYY_.../

where AAA and BBB are any additional parameters that are varied in addition to the six parameters of the baseline model. XXX, YYY, etc encode the data combinations used. These follow the naming conventions described above under Parameter Tables. Each directory contains the main chains, 4-8 text files with one chain in each, and various other files all with names of the form

base_AAA_BBB_XXX_YYY.ext

where ext describes the type of file, and the possible values or ext are

Extension	Data
.txt	parameter chain file with burn in removed
.paramnames	File that describes the parameters included in the chains
.inputparams	Input parameters used when generating the chain
.minimum	Best-fit parameter values, -log likelihoods and chi-square
.minimum.theory_cl	The best-fit temperature and polarization power spectra and lensing potential (see below)
.minimum.plik_foregrounds	The best-fit foreground model (additive component) for each data power spectrum used
.minimum.inputparams	Input parameters used when generating the best fit
.ranges	prior ranges assumed for each parameter

In addition each directory contains any importance sampled outputs with additional data. These have names of the form

base_AAA_BBB_XXX_YYY_post_ZZZ.ext

where ZZZ is the data likelihood that is added by importance sampling. Finally, each directory contains a dist subdirectory, containing results of chain analysis. File names follow the above conventions, with the following extensions

Extension	Data
.margestats	mean, variance and 68, 95 and 99% limits for each parameter (see below)
.likestats	parameters of best-fitting sample in the chain (generally different from the .minmum global best-fit)
.covmat	Covariance matrix for the MCMC parameters
.converge	A summary of various convergence diagnostics

Python scripts for reading in chains and calculating new derived parameter constraints are available as part of CosmoMC, see the readme for details [1]. The config directory in the download includes information about the grid configuration used by the plotting and grid scripts.

File formats

The file formats are standard CosmoMC/GetDist outputs. GetDist includes python scripts for generating tables, 1D, 2D and 3D plots using the provided data, as well as a GUI for conveniently making plots from grid downloads. The formats are summarised here:

Chain files

Each chain file is ASCII and contains one sample on each line. Each line is of the format

weight like param1 param2 param3 …

Here weight is the importance weight or multiplicity count, and like is the total -log Likelihood. param1,param2, etc are the parameter values for the sample, where the numbering is defined by the position in the accompanying .paramnames files.

Note that burn in has been removed from the cosmomc outputs, so full chains provided can be used for analysis. Importance sampled results (with _post) in the name have been thinned by a factor of 10 compared to the original chains, so the files are smaller, but this does not significantly affect the effective number of samples. Note that due to the way MCMC works, the samples in the chain outputs are not independent, but it is safe to use all the samples for estimating posterior averages.

.margestats files

Each row contains the marginalized constraint on individual parameters. The format is fairly self explanatory given the text description in the file, with each line of the form

parameter mean sddev lower1 upper1 limit1 lower2 upper2 limit2 lower3 upper3 limit3

where sddev is the standard deviation, and the limits are 1: 68%, 2: 95%, 3: 99%. The limit tags specify whether a given limit is one tail, two tail or none (if no constraint within the assumed prior boundary).

.minimum.theory_cl files

They contain the best-fit theoretical power spectra (without foregrounds) for each model. The columns are: [math]l[/math], [math]D^{TT}_l[/math], [math]D^{TE}_l[/math], [math]D^{EE}_l[/math], [math]D^{BB}_l[/math], and [math]D^{dd}_l[/math], were [math]D_l \equiv l(l+1) C_l / (2\pi)[/math] in [math]\mu{\rm K}^2[/math]. Also [math]D^{dd}_l= [l(l+1)]^2 C^{\phi\phi}_l/(2\pi)[/math] is the power spectrum of the lensing deflection angle, where [math]C^{\phi\phi}_l[/math] is the lensing potential power spectrum. Note that the lensing spectrum may not be accurate at L > 400 due to the maximum wavenumber and non-linear correction accuracy settings.

Previous Releases: (2015) and (2013) Cosmological Parameters and MC Chains

2015 Release of Cosmological Parameters and MC Chains

Description

The 2015 cosmological parameter results explore a variety of cosmological models with combinations of Planck and other data. We provide results from MCMC exploration chains, as well as best fits, and sets of parameter tables. Definitions, conventions and reference are contained in Planck-2013-XVIPlanck-2015-A15.

Production process

Parameter chains are produced using CosmoMC, a sampling package available here. This includes the sample analysis package (and GUI) GetDist, and the scripts for managing, analysing, and plotting results from the full grid or runs. Chain products provided here have had burn in removed. Some results with additional data are produced by importance sampling.

Note that the baseline model includes one massive neutrino (0.06eV). Grid outputs include WMAP 9 results for consistent assumptions.

Caveats and known issues

1. Confidence intervals are derived from the MCMC samples, and assume the input likelihoods are exactly correct, so there is no quantification for systematic errors other than via the covariance, foreground and beam error models assumed in the likelihood codes.
2. Non-linear lensing modelling uses Halofit; for some extended models and CMB lensing only analyses, tails of the chains may be away from the domain of validity.
3. The CAMB version used for most results is Dec 2014; the Jan 2015 version is used for lensing-only models with neutrinos, and only differs in the neutrino corrections to the Halofit model.
4. There is evidence of temperature-polarization leakage that may affect results including high-L polarization; hence use caution in the interpretation of results including polarization
5. Alternative CamSpec likelihood results in the tables are generated using a slightly older CosmoMC version, with fewer derived parameters and a slightly different BBN predictions for the helium abundance.

Related products

Results of the baseline parameter exploration runs should be reproducible using CosmoMC with the Planck 2015 likelihood codes (when available).

Parameter Tables

These list parameter constraints for each considered model and data combination separately. For the baseline likelihood see

• PDF tables with 68% limits Media:baseline_params_table_2015_limit68.pdf
• PDF tables with 95% limits Media:baseline_params_table_2015_limit95.pdf

There are also larger files including alternative CamSpec and DetSet likelihood results, along with shifts in parameters compared to baseline in units of the baseline error:

• PDF tables with 68% limits Media:params_table_2015_limit68.pdf
• PDF tables with 95% limits Media:params_table_2015_limit95.pdf

Data combination tags used to label results are as follows (see Planck-2013-XVI for full description and references):

Tag	Data
plikHM	baseline high-L Planck power spectra (plik cross half-mission, 30 <= l <= 2508)
plikDS	high-L Planck power spectra (plik cross detsets, 30 <= l <= 2508)
CamSpecHM	high-L Planck power spectra (CamSpec cross half-mission, 30 <= l <= 2500)
CamSpecDS	high-L Planck power spectra (CamSpec cross detsets, 30 <= l <= 2500)
lowl	low-L: Planck temperature only (2 <= l <= 29)
lowTEB	low-L temperature and LFI polarization (2 <= l <= 29)
lowEB	low-L LFI polarization only (2 <= l <= 29)
WMAPTEB	low-L temperature, and LFI+WMAP polarization (2 <= l <= 29)
lensing	Planck lensing power spectrum reconstruction
lensingonly	Planck lensing power spectrum reconstruction only; T,E fixed to best-fit spectrum + priors
BKP	The Bicep-Keck-Planck fiducial B mode likelihood
zre6p5	A hard prior z_re > 6.5
tau07	A Gaussian prior on the optical depth, tau = 0.07 +- 0.02
reion	A hard prior z_re > 6.5, combined with Gaussian prior z_re = 7 +- 1
BAO	Baryon oscillation data from DR11LOWZ, DR11CMASS, MGS and 6DF
JLA	Supernova data from the SDSS-II/SNLS3 Joint Light-curve Analysis
H070p6	Hubble parameter constraint, H_0 = 70.6 +- 3.3
theta	theta_MC fixed to 1.0408
WLonlyHeymans	Conservative cut of the CFHTLenS weak lensing data + priors
WMAP	The full WMAP (temperature and polarization) 9 year data

The high-L Planck likelihoods have TT, TE, EE variants from each spectrum alone, plus the TTTEEE joint constraint.

Tags used to identify the model parameters that are varied are described in Media:parameter_tag_definitions_2015.pdf.

Parameter Chains

We provide the full chains and getdist outputs for our parameter results. The entire grid of results is available from as a 3.7GB compressed file:

• COM_CosmoParams_R2.nn.tar.gz

Where nn is the most recent update. You can also download the Bicep2/Keck/Planck (BKP) joint constraints for +r models, and smaller files containing key results in the base model only:

• COM_CosmoParams_base_plikHM_TT_lowTEB_R2.nn.tar.gz
• COM_CosmoParams_base_plikHM_R2.nn.tar.gz
• COM_CosmoParams_base_lensonly_R2.nn.tar.gz
• COM_CosmoParams_base_r_plikHM_BKP_R2.nn.tar.gz

The download contains a hierarchy of directories, with each separate chain in a separate directory. The structure for the directories is

base_AAA_BBB/XXX_YYY_.../

where AAA and BBB are any additional parameters that are varied in addition to the six parameters of the baseline model. XXX, YYY, etc encode the data combinations used. These follow the naming conventions described above under Parameter Tables. Each directory contains the main chains, 4-8 text files with one chain in each, and various other files all with names of the form

base_AAA_BBB_XXX_YYY.ext

where ext describes the type of file, and the possible values or ext are

Extension	Data
.txt	parameter chain file with burn in removed
.paramnames	File that describes the parameters included in the chains
.inputparams	Input parameters used when generating the chain
.minimum	Best-fit parameter values, -log likelihoods and chi-square
.minimum.theory_cl	The best-fit temperature and polarization power spectra and lensing potential (see below)
.minimum.plik_foregrounds	The best-fit foreground model (additive component) for each data power spectrum used
.minimum.inputparams	Input parameters used when generating the best fit
.ranges	prior ranges assumed for each parameter

In addition each directory contains any importance sampled outputs with additional data. These have names of the form

base_AAA_BBB_XXX_YYY_post_ZZZ.ext

where ZZZ is the data likelihood that is added by importance sampling. Finally, each directory contains a dist subdirectory, containing results of chain analysis. File names follow the above conventions, with the following extensions

Extension	Data
.margestats	mean, variance and 68, 95 and 99% limits for each parameter (see below)
.likestats	parameters of best-fitting sample in the chain (generally different from the .minmum global best-fit)
.covmat	Covariance matrix for the MCMC parameters
.converge	A summary of various convergence diagnostics

Python scripts for reading in chains and calculating new derived parameter constraints are available as part of CosmoMC, see the readme for details [2]. The config directory in the download includes information about the grid configuration used by the plotting and grid scripts.

File formats

The file formats are standard CosmoMC/GetDist outputs. GetDist includes python scripts for generating tables, 1D, 2D and 3D plots using the provided data, as well as a GUI for conveniently making plots from grid downloads. The formats are summarised here:

Chain files

Each chain file is ASCII and contains one sample on each line. Each line is of the format

weight like param1 param2 param3 …

Here weight is the importance weight or multiplicity count, and like is the total -log Likelihood. param1,param2, etc are the parameter values for the sample, where the numbering is defined by the position in the accompanying .paramnames files.

Note that burn in has been removed from the cosmomc outputs, so full chains provided can be used for analysis. Importance sampled results (with _post) in the name have been thinned by a factor of 10 compared to the original chains, so the files are smaller, but this does not significantly affect the effective number of samples. Note that due to the way MCMC works, the samples in the chain outputs are not independent, but it is safe to use all the samples for estimating posterior averages.

.margestats files

Each row contains the marginalized constraint on individual parameters. The format is fairly self explanatory given the text description in the file, with each line of the form

parameter mean sddev lower1 upper1 limit1 lower2 upper2 limit2 lower3 upper3 limit3

where sddev is the standard deviation, and the limits are 1: 68%, 2: 95%, 3: 99%. The limit tags specify whether a given limit is one tail, two tail or none (if no constraint within the assumed prior boundary).

.minimum.theory_cl files

They contain the best-fit theoretical power spectra (without foregrounds) for each model. The columns are: [math]l[/math], [math]D^{TT}_l[/math], [math]D^{TE}_l[/math], [math]D^{EE}_l[/math], [math]D^{BB}_l[/math], and [math]D^{dd}_l[/math], were [math]D_l \equiv l(l+1) C_l / (2\pi)[/math] in [math]\mu{\rm K}^2[/math]. Also [math]D^{dd}_l= [l(l+1)]^2 C^{\phi\phi}_l/(2\pi)[/math] is the power spectrum of the lensing deflection angle, where [math]C^{\phi\phi}_l[/math] is the lensing potential power spectrum. Note that the lensing spectrum may not be accurate at L > 400 due to the maximum wavenumber and non-linear correction accuracy settings.

2013 Release of Cosmological parameters and MC Chains

Description

The cosmological parameter results explore a variety of cosmological models with combinations of Planck and other data. We provide results from MCMC exploration chains, as well as best fits, and sets of parameter tables. Definitions, conventions and reference are contained in Planck-2013-XVI.

Production process

Parameter chains are produced using CosmoMC, a sampling package available here. This includes the sample analysis package GetDist, and the scripts for managing, analysing, and plotting results from the full grid or runs. Chain products provided here have had burn in removed. Some results with additional data are produced by importance sampling.

Note that the baseline model includes one massive neutrino (0.06eV). Grid outputs include WMAP 9 results for consistent assumptions.

Caveats and known issues

1. Confidence intervals are derived from the MCMC samples, and assume the input likelihoods are exactly correct, so there is no quantification for systematic errors other than via the covariance, foreground and beam error models assumed in the likelihood codes. We had some issues producing reliable results from the minimizer used to produce the best fits, so in some cases the quoted fits may be significantly improved. The chain outputs contain some parameters that are not used, for example the beam mode ranges for all but the first mode (the beam modes are marginalised over anlaytically internally to the likelihood).
2. Where determined from BBN consistency, the [math]Y_P[/math] parameter uses an interpolation table from based on the 2008 version of the Parthenope BBN code. More recent updates to the neutron lifetime suggest that the [math]Y_P[/math] values reported in the tables may be in error by around 0.0005. This has a negligible impact on the predicted CMB power spectrum or any of the parameter results reported in this series of papers. However, the difference should be taken into account when comparing with BBN results reported in Sect. 6.4. of Planck-2013-XVI, which use an updated version for the neutron lifetime (and several other nuclear reaction rates that have negligible impact). Note also that the error on [math]Y_P[/math] quoted in the tables here does not include theoretical errors in the BBN prediction.

Related products

Results of the parameter exploration runs should be reproducible using CosmoMC with the Planck likelihood code.

Parameter Tables

These list paramter constraints for each considered model and data combination separately

• PDF tables with 68% limits File:Grid limit68.pdf
• PDF tables with 95% limits File:Grid limit95.pdf

There are also summary comparison tables, showing how constraints for selected models vary with data used to constrain them:

• Comparison tables with 68% limits File:Comparetables limit68.pdf
• Comparison tables with 95% limits File:Comparetables limit95.pdf

Data combination tags used to label results are as follows (see Planck-2013-XVI for full description and references):

Tag	Data
planck	high-L Planck temperature (CamSpec, 50 <= l <= 2500)
lowl	low-L: Planck temperature (2 <= l <= 49)
lensing	Planck lensing power spectrum reconstruction
lowLike	low-L WMAP 9 polarization (WP)
tauprior	A Gaussian prior on the optical depth, tau = 0.09 +- 0.013
BAO	Baryon oscillation data from DR7, DR9 and and 6DF
SNLS	Supernova data from the Supernova Legacy Survey
Union2	Supernova data from the Union compilation
HST	Hubble parameter constraint from HST (Riess et al)
WMAP	The full WMAP (temperature and polarization) 9 year data

Tags used to identify the model paramters that are varied are described in File:Parameter tag definitions.pdf. Note that alpha1 results are not used in the parameter paper, and are separate from the isocurvature results in the inflation paper.

Parameter Chains

We provide the full chains and getdist outputs for our parameter results. The entire grid of results is available from as a 2.8GB compressed file:

• Full Grid Download

You can also download key chains for the baseline LCDM model here:

• Planck+lensing
• Planck+WP
• Planck+WP+lensing
• Planck+WP+highL
• Planck+WP+highL+lensing

The download contains a hierarchy of directories, with each separate chain in a separate directory. The structure for the directories is

base_AAA_BBB/XXX_YYY_.../

where AAA and BBB are any additional parameters that are varied in addition to the six parameters of the baseline model. XXX, YYY, etc encode the data combinations used. These follow the naming conventions described above under Parameter Tables. Each directory contains the main chains, 4-8 text files with one chain in each, and various other files all with names of the form

base_AAA_BBB_XXX_YYY.ext

where ext describes the type of file, and the possible values or ext are

Extension	Data
.txt	parameter chain file with burn in removed
.paramnames	File that describes the parameters included in the chains
.minimum	Best-fit parameter values, -log likelihoods and chi-square
.bestfit_cl	The best-fit temperature and polarization power spectra and lensing potential (see below)
.inputparams	Input parameters used when generating the chain
.minimum.inputparams	Input parameters used when generating the best fit
.ranges	prior ranges assumed for each parameter

In addition each directory contains any importanced sampled outputs with additional data. These have names of the form

base_AAA_BBB_XXX_YYY_post_ZZZ.ext

where ZZZ is the data likelihood that is added by importance sampling. Finally, each directory contains a dist subdirectory, containing results of chain analysis. File names follow the above convntions, with the following extensions

Extension	Data
.margestats	mean, variance and 68, 95 and 99% limits for each parameter (see below)
.likestats	parameters of best-fitting sample in the chain (generally different from the .minmum global best-fit)
.covmat	Covariance matrix for the MCMC parameters
.corr	Correlation matrix for the parameters
.converge	A summary of various convergence diagnostics

Python scripts for reading in chains and calculating new derived parameter constraints are available as part of CosmoMC, see the readme for details [3].

File formats

The file formats are standard March 2013 CosmoMC outputs. CosmoMC includes python scripts for generating tables, 1D, 2D and 3D plots using the provided data. The formats are summarised here:

Chain files

Each chain file is ASCII and contains one sample on each line. Each line is of the format

weight like param1 param2 param3 …

Here weight is the importance weight or multiplicity count, and like is the total -log Likelihood. param1,param2, etc are the parameter values for the sample, where the numbering is defined by the position in the accompanying.paramnames files.

Note that burn in has been removed from the cosmomc outputs, so full chains provided can be used for analysis. Importance sampled results (with _post) in the name have been thinned by a factor of 10 compared to the original chains, so the files are smaller, but this does not significantly affect the effective number of samples. Note that due to the way MCMC works, the samples in the chain outputs are not independent, but it is safe to use all the samples for estimating posterior averages.

.margestats files

Each row contains the marginalized constraint on individual parameters. The format is fairly self explanatory given the text description in the file, with each line of the form

parameter mean sddev lower1 upper1 limit1 lower2 upper2 limit2 lower3 upper3 limit3

where sddev is the standard deviation, and the limits are 1: 68%, 2: 95%, 3: 99%. The limit tags specify whether a given limit is one tail, two tail or none (if no constraint within the assumed prior boundary).

.bestfit_cl files

They contain the best-fit theoretical power spectra (without foregrounds) for each model. The columns are: [math]l[/math], [math]D^{TT}_l[/math], [math]D^{TE}_l[/math], [math]D^{EE}_l[/math], [math]D^{BB}_l[/math], and [math]D^{dd}_l[/math], were [math]D_l \equiv l(l+1) C_l / (2\pi)[/math] in [math]\mu{\rm K}^2[/math]. Also [math]D^{dd}_l= [l(l+1)]^2 C^{\phi\phi}_l/(2\pi)[/math] is the power spectrum of the lensing deflection angle, where [math]C^{\phi\phi}_l[/math] is the lensing potential power spectrum. For results not including the lensing likelihood, this is the prediction from linear theory; for lensing outputs this includes corrections due to non-linear structure growth. The [math]D_l[/math] are output to high [math]l[/math], but not actually computed above [math]l_{\rm max}=2500[/math] (Planck), [math]l_{\rm max}=4500[/math] (Planck+highL) or [math]l_{\rm max}=1500[/math] (WMAP), and [math]l[/math] values above these are fixed to a scaled fiducial template.