Difference between revisions of "CMB spectrum & Likelihood Code"

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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 {{PlanckPapers|planck2013-p08}}. 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 {{PlanckPapers|planck2014-p15}} and in {{PlanckPapers|planck2014-p13}}. 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 {{PlanckPapers|planck2014-p15}}). A thorough description of the models of unresolved foregrounds is given in {{PlanckPapers|planck2014-p13}}. 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:
 
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 {{PlanckPapers|planck2013-p08}}. 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 {{PlanckPapers|planck2014-p15}} and in {{PlanckPapers|planck2014-p13}}. 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 {{PlanckPapers|planck2014-p15}}). A thorough description of the models of unresolved foregrounds is given in {{PlanckPapers|planck2014-p13}}. 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:
 
#Unbinned, with 2479 bandpowers (<math>\ell=30-2508</math>).
 
#Unbinned, with 2479 bandpowers (<math>\ell=30-2508</math>).
#Binned, in bins of <math> \Delta\ell=30 </math>, with 83 bandpowers in total. We bin the <math> C_\ell </math> power spectrum with a weight proportional to <math> \ell (\ell+1) </math>, so that the <math> C_{\ell_b} </math> binned bandpower centered in <math> \ell_b </math> is: <math> \\ C_{\ell_b}=\Sigma_{\ell \in b}  w_{\ell_b\ell} C_\ell \quad \text{with} \quad w_{\ell_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_b\times n_\ell</math> matrix, with <math>n_b=83</math> the number of bins and <math>n_\ell=2479</math> the number of unbinned multipoles. Thus:  <math> \\ \vec{C_\mathrm{binned}}=B \, \vec{C} \\ \mathrm{cov_\mathrm{binned}}= B\, \mathrm{cov}\, B^T \\ \ell_b=B\, \ell \\ </math> Here, <math> \vec{C_{binned}} (\vec{C}) </math>  is the vector containing all the binned (unbinned) <math>C_\ell</math> bandpowers, <math>\mathcal{C} </math> is the covariance matrix and <math>\ell_b</math> is  the weighted multipole average in each bin. The binned <math>D_{\ell_B}</math> power spectrum is then calculated as: <math> \\ D_{\ell_b}=\ell_b (\ell_b+1)/2/\pi C_{\ell_b} </math>.
+
#Binned, in bins of <math> \Delta\ell=30 </math>, with 83 bandpowers in total. We bin the <math> C_\ell </math> power spectrum with a weight proportional to <math> \ell (\ell+1) </math>, so that the <math> C_{\ell_b} </math> binned bandpower centered in <math> \ell_b </math> is: <math> \\ C_{\ell_b}=\Sigma_{\ell \in b}  w_{\ell_b\ell} C_\ell \quad \text{with} \quad w_{\ell_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_b\times n_\ell</math> matrix, with <math>n_b=83</math> the number of bins and <math>n_\ell=2479</math> the number of unbinned multipoles. Thus:  <math> \\ \vec{C}_\mathrm{binned}=B \, \vec{C} \\ \mathrm{cov_\mathrm{binned}}= B\, \mathrm{cov}\, B^T \\ \ell_b=B\, \ell \\ </math> Here, <math> \vec{C}_{binned} (\vec{C}) </math>  is the vector containing all the binned (unbinned) <math>C_\ell</math> bandpowers, <math>\mathrm{cov} </math> is the covariance matrix and <math>\ell_b</math> is  the weighted multipole average in each bin. The binned <math>D_{\ell_B}</math> power spectrum is then calculated as: <math> \\ D_{\ell_b}=\ell_b (\ell_b+1)/2/\pi C_{\ell_b} </math>.
 +
 
 
===Inputs===
 
===Inputs===
  
 
+
; Low-l spectrum (<math>\ell < 30</math>):
; Low-l spectrum (<math>\ell < 50</math>):
 
 
* frequency maps from 30–353 GHz
 
* frequency maps from 30–353 GHz
 
* common mask {{PlanckPapers|planck2013-p06}}
 
* common mask {{PlanckPapers|planck2013-p06}}
 
* compact sources catalog
 
* compact sources catalog
  
; High-l spectrum (<math>50 < \ell < 2500</math>):  
+
; High-l spectrum (<math>30 < \ell < 2500</math>):  
 
   
 
   
* 100, 143, 143x217 and 217 GHz spectra and their covariance matrix (Sec. 2 in {{PlanckPapers|planck2013-p08}})
+
* 100, 143, 143x217 and 217 GHz spectra and their covariance matrix (Sec. 3.3 {{PlanckPapers|planck2014-p15}})
* best-fit foreground templates and inter-frequency calibration factors (Table 5 of {{PlanckPapers|planck2013-p11}})
+
* best-fit foreground templates and inter-frequency calibration factors (Table 3 of {{PlanckPapers|planck2014-p15}})
* Beam transfer function uncertainties {{PlanckPapers|planck2013-p03c}}
+
* Beam transfer function uncertainties {{PlanckPapers|planck2014-p08}}
 
=== File names and Meta data ===
 
=== File names and Meta data ===
  
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* ''{{PLASingleFile | fileType=cosmo | name=COM_PowerSpect_CMB_R1.10.fits | link=COM_PowerSpect_CMB_R1.10.fits}}''  
 
* ''{{PLASingleFile | fileType=cosmo | name=COM_PowerSpect_CMB_R1.10.fits | link=COM_PowerSpect_CMB_R1.10.fits}}''  
  
which contains 3 extensions
+
which contains 5 extensions
  
 
; LOW-ELL (BINTABLE)
 
; LOW-ELL (BINTABLE)
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# ''ERRDOWN'' (float): the downward uncertainty
 
# ''ERRDOWN'' (float): the downward uncertainty
  
; HIGH-ELL (BINTABLE)
+
; 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:
+
: with the high-ell part of the spectrum, binned into 83 bins covering <math>\langle l \rangle = 47-2499\ </math> in bins of width <math>l=30</math> (with the exception of the last bin that is smaller). The table columns are as follows:
 
# ''ELL'' (integer): mean multipole number of bin
 
# ''ELL'' (integer): mean multipole number of bin
 
# ''L_MIN'' (integer): lowest multipole of bin
 
# ''L_MIN'' (integer): lowest multipole of bin
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; COV-MAT (IMAGE)
 
; 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.
+
: 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.
  
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.
 
  
[[File: CMBspect.jpg|thumb|center|700px|'''CMB spectrum. Linear x-scale; error bars only at high <math>\ell</math>.''']]
+
; HIGH-ELL (BINTABLE)
 +
: with the high-ell part of the spectrum, unbinned, in 2979 bins covering <math>\langle l \rangle = 30-2508\ </math>. The table columns are as follows:
 +
# ''ELL'' (integer):  multipole
 +
# ''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 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:
 
The CMB spectrum is also given in a simple text comma-separated file:

Revision as of 00:45, 4 February 2015


CMB spectra[edit]

General description[edit]

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[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 [math]\ell=30[/math], 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 [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[3]).

Production process[edit]

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[1]. 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[4]. 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[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 [5] and in [6]. 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 [5]). A thorough description of the models of unresolved foregrounds is given in [6]. 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 [math] C_\ell [/math] power spectrum with a weight proportional to [math] \ell (\ell+1) [/math], so that the [math] C_{\ell_b} [/math] binned bandpower centered in [math] \ell_b [/math] is: [math] \\ C_{\ell_b}=\Sigma_{\ell \in b} w_{\ell_b\ell} C_\ell \quad \text{with} \quad w_{\ell_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_b\times n_\ell[/math] matrix, with [math]n_b=83[/math] the number of bins and [math]n_\ell=2479[/math] the number of unbinned multipoles. Thus: [math] \\ \vec{C}_\mathrm{binned}=B \, \vec{C} \\ \mathrm{cov_\mathrm{binned}}= B\, \mathrm{cov}\, B^T \\ \ell_b=B\, \ell \\ [/math] Here, [math] \vec{C}_{binned} (\vec{C}) [/math] is the vector containing all the binned (unbinned) [math]C_\ell[/math] bandpowers, [math]\mathrm{cov} [/math] is the covariance matrix and [math]\ell_b[/math] is the weighted multipole average in each bin. The binned [math]D_{\ell_B}[/math] power spectrum is then calculated as: [math] \\ D_{\ell_b}=\ell_b (\ell_b+1)/2/\pi C_{\ell_b} [/math].

Inputs[edit]

Low-l spectrum ([math]\ell \lt 30[/math])
High-l spectrum ([math]30 \lt \ell \lt 2500[/math])
  • 100, 143, 143x217 and 217 GHz spectra and their covariance matrix (Sec. 3.3 [5])
  • best-fit foreground templates and inter-frequency calibration factors (Table 3 of [5])
  • Beam transfer function uncertainties [7]

File names and Meta data[edit]

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 [math]\langle l \rangle = 47-2499\ [/math] in bins of width [math]l=30[/math] (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.


HIGH-ELL (BINTABLE)
with the high-ell part of the spectrum, unbinned, in 2979 bins covering [math]\langle l \rangle = 30-2508\ [/math]. 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:

Likelihood[edit]

TO BE WRITTEN.

References[edit]

  1. 1.01.11.2 Planck 2013 results. XI. Component separation, Planck Collaboration, 2014, A&A, 571, A11
  2. Planck 2015 results. XI. CMB power spectra, likelihoods, and robustness of cosmological parameters, Planck Collaboration, 2016, A&A, 594, A11.
  3. Planck 2015 results. XIII. Cosmological parameters, Planck Collaboration, 2016, A&A, 594, A13.
  4. 4.04.1 Planck 2013 results. XV. CMB power spectra and likelihood, Planck Collaboration, 2014, A&A, 571, A15
  5. 5.05.15.25.3
  6. 6.06.1

Cosmic Microwave background

Flexible Image Transfer Specification