Cosmological Parameters

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 [1].

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.