CMB spectrum & Likelihood Code

CMB spectra

General description

The Planck best-fit CMB temperature power spectrum, shown in figure below, covers the wide range of multipoles [math] \ell [/math] = 2-2508. Over the multipole range [math] \ell [/math] = 2–29, the power spectrum is derived from a component-separation algorithm, Commander, UPDATE COMMANDER: applied to maps in the frequency range 30–353 GHz over 91% of the sky Planck-2013-XII . 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 [math]\ell=30[/math], instead, the spectrum is derived from the Plik likelihood Planck-2015-A11 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 [math]\ell=30[/math], linear at higher [math]\ell[/math]; 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).

Production process

UPDATE COMMANDER The [math]\ell[/math] < 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. The power spectrum at any multipole [math]\ell[/math] is given as the maximum probability point for the posterior [math]C_\ell[/math] distribution, marginalized over the other multipoles, and the error bars are 68% confidence level Planck-2013-XV. The [math]\ell[/math] > 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. 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 and in . 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 ). A thorough description of the models of unresolved foregrounds is given in . The spectrum covariance matrix accounts for cosmic variance and noise contributions, together with beam uncertainties. The [math]\ell[/math] > 30 CMB TT spectrum and associated covariance matrix are available in two formats:

1. Unbinned, with 2479 bandpowers ([math]\ell=30-2508[/math]).
2. Binned, in bins of [math]\Delta\ell=30 [/math], with 83 bandpowers in total. We bin the Cl power spectra with a weight proportional to [math] \ell (\ell+1) [/math], so that the binned bandpower centered in [math]\ell_b[/math] is: [math] \\ C_{\ell_b}=\Sigma_{\ell \in b} w^b_\ell C_\ell \quad \text{with} \quad w^b_\ell=\frac{\ell (\ell+1)}{\Sigma_{\ell \in b} \ell (\ell+1)}. \\[/math] Equivalently, using the matrix formalism, we can construct the binning matrix B as: [math]\\ B_{\ell \ell_b}=w^{\ell_b}_\ell \\ [/math] where B is a [math] n_bxn_\ell=83x2479[/math] matrix, with [math]n_b\lt math\gt the number of bins and \lt math\gt n_\ell\lt math\gt the number of unbinned multipoles. Thus: \lt math\gt \\ \vec{C}^{binned}_{\ell_b}=B \vec{C}_\ell \\ cov^{binned}= B cov B^T \\ \ell_b=B \ell \\ [/math] Here, \vec{C}^{binned}_{\ell_b} (\vec{C}_{\ell}) just indicates the vector containing all the binned (unbinned) bandpowers. Note that we use the binning matrix also to calculate the average value of the bin [math]\ell_b\lt math\gt . The binned \lt math\gt D_{\ell_B}\lt math\gt power spectra are then calculated as: \lt math\gt \\ D_\ell_b=\ell_b (\ell_b+1)/2/\pi C_{\ell_b} \\ [/math].

Inputs

Low-l spectrum ([math]\ell \lt 50[/math])

• frequency maps from 30–353 GHz
• common mask Planck-2013-XII
• compact sources catalog

High-l spectrum ([math]50 \lt \ell \lt 2500[/math])

• 100, 143, 143x217 and 217 GHz spectra and their covariance matrix (Sec. 2 in Planck-2013-XV)
• best-fit foreground templates and inter-frequency calibration factors (Table 5 of Planck-2013-XVI)
• Beam transfer function uncertainties Planck-2013-VII

File names and Meta data

The CMB spectrum and its covariance matrix are 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

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 74 bins covering [math]\langle l \rangle = 47-2419\ [/math] in bins of width [math]l=31[/math] (with the exception of the last 4 bins that are wider). 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 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-[math]\ell[/math] parts, respectively, and with the error bars for the high-ell part only in order to avoid confusion.

CMB spectrum. Linear x-scale; error bars only at high [math]\ell[/math].

The CMB spectrum is also given in a simple text comma-separated file:

• COM_PowerSpect_CMB_R1.10.txt

Likelihood

TO BE WRITTEN.