# CMB spectrum and likelihood code

## Contents

## General description[edit]

### CMB spectra[edit]

The Planck best-fit CMB temperature power spectrum, shown in figure below, covers the wide range of multipoles . Over the multipole range , 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 #planck2013-p06. 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 #planck2013-p08 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.

### Likelihood[edit]

The likelihood code and data allow to compute the likelihood of a model that predicts the CMB power spectra, lensing power spectrum and foreground and some instrumental parameters. The data file are built from the Planck mission results, as well as the some ancillary data from the wmap9 data release. The data file are in a specific internal format and can only be read by the code. The code consists in a c/f90 library, along with some optional tools in python. The code allows to read the data files, and provided model power spectra and nuisance parameters to compute the log likelihood of the model.

Detailed description of the installation and usage of the likelihood code and data is provided in the package. The package includes six data files: five for the CMB likelihoods and one for the lensing likelihood. All of the likelihood delivered are described full in the Power spectrum & Likelihood Paper #planck2013-p08 (for the CMB based likelihood) and in the Lensing Paper (for the lensing likelihood) #planck2013-p12.

The CMB full likelihood has been cut in 3 different part to allow using selectively different range of multipoles. It also reflects the fact that the mathematical approximation used for those different part are very different, as well as the underlying data. In details, we are distributing one low- Temperature only likelihood (commander), one low- Temperature and Polarisation likelihood (lowlike) and one higl- likelihood CAMspec.

The commander likelihood is covering 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 do the likelihood computation in the code. See #planck2013-p08 section 8.1 for more details.The lowlike likelihood is covering the multipole 2 to 32 for Temperature and Polarization data. Planck is not releasing any polarisation data in this release. We are using here the WMAP9 polarization map which are included in the data package. A temperature map is needed to perform the computation nevertheless, and we are using 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 is essentially providing 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 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 #planck2013-p08 section 8.3 for more details. Note that the version of the WMAP code we are currently using (code version v1.0) does not perform any test on the positive definiteness of the TT, TE, EE covariance matrix, 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 is covering the multipoles 50 to 2500 for Temperature. 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 217Ghz channels. Doing so it must model the foreground in each of those frequency using a 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. Detailled description of the different nuisance parameter names and meaning is given below. Priors are included in the likelihood on the cib spectral index, relative calibration factors and beam error eigenmodes. See #planck2013-p08 section 2.1 for more details.

The actspt likelihood covers the multipole 1500 to 10000 for Temperature. It is described in #dun2013,#Keis2011,#Reic2012. It uses the code and data that can be retrieved here. It has been slightly modified to use a thermal and kinetci SZ model that matches the one used in CAMspec. As stated in #dun2013, the dust parameters a_ge and a_gs must be explored witht he 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 is covering the multipoles 40 to 400. It uses 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 one 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 #planck2013-p12 section 6.1 for more details.

## Production process[edit]

### CMB spectra[edit]

The

< 50 part of the Planck 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 #planck2013-p06. 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% CL #planck2013-p08.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 #planck2013-p08. 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 #planck2013-p08. 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 #planck2013-p11). A thorough description of the models of unresolved foregrounds is given in Sec. 3 of #planck2013-p08 and Sec. 4 of #planck2013-p11. 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.

> 50 part of the### Likelihood[edit]

The code is based upon some basic routine from the libpmc library in the cosmoPMC code. It also uses some code from the WMAP9 likelihood for the lowlike likelihood and #dun2013,#Keis2011,#Reic2012 for the actspt 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 #planck2013-p08 (section 2 and 8) and in the lensing paper #planck2013-p12 (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[edit]

### CMB spectra[edit]

- Low-l spectrum ( )

- frequency maps from 30–353 GHz;
- common mask #planck2013-p06;
- compact sources catalog.

- High-l spectrum ( )

- 100, 143, 143x217 and 217 GHz spectra and their covariance matrix (Sec. 2 in #planck2013-p08);
- best-fit foreground templates and inter-frequency calibration factors (Table 5 of #planck2013-p11);
- Beam transfer function uncertainties #planck2013-p03c;

### Likelihood[edit]

- commander
- All Planck channels maps, compact source catalogs, common masks, beam transfer functions for all channels.

- lowlike
- WMAP9 likelihood data. Low-ell commander map.

- CAMspec
- 100,143 & 217GHz detector and detests maps. 857GHz chanel Map. compact source catalog. Common masks (0,1 & 3). beam transfer function and error eigenmodes and covariance for 100,143 and 217Ghz detectors & detsets. Theoretical templates for the tSZ and kSZ contributions. Color corrections for the CIB emission for the 143GHz and 217GHz detectors & detsets. Fiducial CMB model (bootstrapped from WMAP7 best fit spectrum) estimated noise contribution from the half-ring maps for 100, 143 & 217GHz.

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

- act/spt
- The data and code from here. The TSZ and KSZ template are changed to match those of CAMspec.

## File names and Meta data[edit]

### CMB spectra[edit]

The CMB spectrum and its covariance matrix is distributed in a single FITS file named *COM_PowerSpect_CMB_R1.10.fits* 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 $\langle l \rangle = 47-2419$ in bins of width $l=31$ (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[edit]

- the source code is in the file
- COM_Code_Likelihood-v1.0_R1.10.tar.gz(C, f90 and python likelihood library and tools)

- 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 of the package comes with a README file describing the full package. 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 modelling granted 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 into 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.

- The masks used in the Likelihood paper #planck2013-p08 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 |

### Retrieval from the Planck Legacy Archive[edit]

The CMB spectra and likelihood files can be retrieved from the Planck Legacy Archive. One should select "Cosmology products" and look at the "CMB angular power spectra" and "Likelihood" sections. The files can be downloaded directly or through the "Shopping Basket".

## References[edit]

<biblio force=false>

</biblio>

Cosmic Microwave background

Sunyaev-Zel'dovich

Flexible Image Transfer Specification

(Planck) Low Frequency Instrument

(Planck) High Frequency Instrument