# 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).
- Where determined from BBN consistency, the 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 #planck2013-p11, 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 quoted in the tables here does not include theoretical errors in the BBN prediction. parameter uses an interpolation table from #Hamann2007sb based on the 2008 version of the Parthenope BBN code. More recent updates to the neutron lifetime suggest that the values reported in the tables may be in error by around 0.0005. This has a negligible impact on the predicted

## 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

Data combination tags used to label results are as follows (see #planck2013-p11 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[edit]

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:

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

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 [2].

## 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 (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: $l$, $D^{TT}_l$, $D^{TE}_l$, $D^{EE}_l$, $D^{BB}_l$, and $D^{dd}_l$, were $D_l \equiv l(l+1) C_l / (2\pi)$ in $\mu{\rm K}^2$. Also $D^{dd}_l= [l(l+1)]^2 C^{\phi\phi}_l/(2\pi)$ is the power spectrum of the lensing deflection angle, where $C^{\phi\phi}_l$ 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 $l_{\rm max}=2500$ (Planck), $l_{\rm max}=4500$ (Planck+highL) or $l_{\rm max}=1500$ (WMAP), and $l$ values above these are fixed to a scaled fiducial template.

## References[edit]

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Cosmic Microwave background