# Cosmological Parameters

## Contents

## Description[edit]

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 #planck2013-p11.

## Production process[edit]

Parameter chains are produced using CosmoMC, a sampling package available from [1]. 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[edit]

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).

## Related products[edit]

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

## Parameter Tables[edit]

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

## Parameter Chains[edit]

We provide the full chains and getdist outputs for our parameter results. The entire grid of results is available from the PLA (link below) as a 2.8GB compressed file. You can also download key chains for the baseline LCDM model here:

Baseline LCDM model chains |
---|

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 variend in addition to the six parameters of the baseline model. XXX, YYY, etc encode the data combinations used. The parameter tags are defined in #planck2013-p11. Data combination tags are as follows (see the parameters paper 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 |

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

Tag | 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 importances 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.

In addition to the main ouputs, each directory contains a *dist* subdirectory, containing results of chain analysis. File names follow the above convntions, with the following extensions

Tag | 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 |

## File formats[edit]

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 (wth *_post*) in the name have been thinnen 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 indpendent, 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 constaint within the assumed prior boundary).

- .bestfit_cl files
- The contain the best-fit theoretical power spectra (without foregrounds) for each model. The columns are

l D^TT_l D^TE_l D^EE_l D^BB_l C^dd_l

The D_l's are all l(l+1) C_l / (2pi)'s in (microK)^2.

C^dd_l= (l(l+1))^2*C^Phi_l/(2pi) is the power spectrum of the lensing deflection angle, where C^Phi_l 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 D_l are output to high L, but not actually computed above Lmax=2500 (Planck), Lmax=4500 (Planck+highL) or Lmax=1500 (WMAP), and L values above these are fixed to a scaled fiducial template.

## References[edit]

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Planck Legacy Archive