# CMB spectrum and likelihood code

## CMB spectra

### General description

The Planck best-fit CMB temperature power spectrum, shown in figure below, covers the wide range of multipoles = 2-2508. UPDATE COMMANDER: Over the multipole range = 2–29, 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 Planck-2013-XII[1] . The asymmetric error bars associated to this spectrum are the 68% confidence limits and include the uncertainties due to foreground subtraction.

For multipoles equal or greater than , instead, the spectrum is derived from the Plik likelihood Planck-2015-A11[2] by optimally combining the spectra in the frequency range 100-217 GHz, and correcting them for unresolved foregrounds. Associated 1-sigma errors include beam uncertainties. Both Commander and Plik are described in more details in the sections below.

CMB spectrum. Logarithmic x-scale up to , linear at higher ; all points with error bars. The red line is the Planck best-fit primordial power spectrum (cf Planck TT+lowP in Table 3 of Planck-2015-A13[3]). The blue shaded area shows the uncertainties due to cosmic variance alone.

### Production process

UPDATE COMMANDER 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 Planck-2013-XII[1]. 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% confidence level Planck-2013-XV[4].

The > 30 part of the CMB temperature power spectrum has been derived by the Plik likelihood, a code that implements a pseudo-Cl based technique, extensively described in Sec. 2 and the Appendix of Planck-2013-XV[4]. Frequency spectra are computed as cross-spectra between half-mission maps. Mask and multipole range choices for each frequency spectrum are summarized in Section 3.3 of Planck-2015-A13[3] and in Planck-2015-A11[2]. 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+TT+lowP in Table 3 of Planck-2015-A13[3]). A thorough description of the models of unresolved foregrounds is given in Planck-2015-A11[2]. The spectrum covariance matrix accounts for cosmic variance and noise contributions, together with beam uncertainties. The > 30 CMB TT spectrum and associated covariance matrix are available in two formats:

1. Unbinned, with 2479 bandpowers ().
2. Binned, in bins of , with 83 bandpowers in total. We bin the power spectrum with a weight proportional to , so that the binned bandpower centered in is: Equivalently, using the matrix formalism, we can construct the binning matrix B as: where B is a matrix, with the number of bins and the number of unbinned multipoles. Thus: Here, is the vector containing all the binned (unbinned) bandpowers, is the covariance matrix and is the weighted multipole average in each bin. The binned power spectrum is then calculated as: .

### Inputs

UPDATE COMMANDER

Low-l spectrum ()
High-l spectrum ()

### File names and Meta data

CHECK EXTENSION NAMES

The CMB spectrum and its covariance matrix are distributed in a single FITS file named

which contains 5 extensions

LOW-ELL (BINTABLE)
with the low ell part of the spectrum, not binned, and for l=2-49. The table columns are
1. ELL (integer): multipole number
2. D_ELL (float): $D_l$ as described below
3. ERRUP (float): the upward uncertainty
4. ERRDOWN (float): the downward uncertainty
HIGH-ELL (BINTABLE)
with the high-ell part of the spectrum, binned into 83 bins covering in bins of width (with the exception of the last bin that is smaller). The table columns are as follows:
1. ELL (integer): mean multipole number of bin
2. L_MIN (integer): lowest multipole of bin
3. L_MAX (integer): highest multipole of bin
4. D_ELL (float): $D_l$ as described below
5. ERR (float): the uncertainty
COV-MAT (IMAGE)
with the covariance matrix of the high-ell part of the spectrum in a 83x83 pixel image, i.e., covering the same bins as the HIGH-ELL table.

Note that this is the covariance matrix of the , not of the .

HIGH-ELL (BINTABLE)
with the high-ell part of the spectrum, unbinned, in 2979 bins covering . The table columns are as follows:
1. ELL (integer): multipole
2. D_ELL (float): $D_l$ as described below
3. ERR (float): the uncertainty
COV-MAT (IMAGE)
with the covariance matrix of the high-ell part of the spectrum in a 2979x2979 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 CMB spectrum is also given in a simple text comma-separated file:

TO BE WRITTEN.

## References

Cosmic Microwave background

Flexible Image Transfer Specification